TSTP Solution File: LCL550+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : LCL550+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 18 04:56:39 EDT 2022
% Result : Theorem 4.30s 3.00s
% Output : Proof 4.37s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL550+1 : TPTP v8.1.0. Released v3.3.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n021.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Sep 1 22:10:28 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.35 Usage: tptp [options] [-file:]file
% 0.12/0.35 -h, -? prints this message.
% 0.12/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.35 -m, -model generate model.
% 0.12/0.35 -p, -proof generate proof.
% 0.12/0.35 -c, -core generate unsat core of named formulas.
% 0.12/0.35 -st, -statistics display statistics.
% 0.12/0.35 -t:timeout set timeout (in second).
% 0.12/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.35 -<param>:<value> configuration parameter and value.
% 0.12/0.35 -o:<output-file> file to place output in.
% 4.30/3.00 % SZS status Theorem
% 4.30/3.00 % SZS output start Proof
% 4.30/3.00 tff(is_a_theorem_type, type, (
% 4.30/3.00 is_a_theorem: $i > $o)).
% 4.30/3.00 tff(strict_implies_type, type, (
% 4.30/3.00 strict_implies: ( $i * $i ) > $i)).
% 4.30/3.00 tff(tptp_fun_Y_3_type, type, (
% 4.30/3.00 tptp_fun_Y_3: $i)).
% 4.30/3.00 tff(and_type, type, (
% 4.30/3.00 and: ( $i * $i ) > $i)).
% 4.30/3.00 tff(tptp_fun_X_4_type, type, (
% 4.30/3.00 tptp_fun_X_4: $i)).
% 4.30/3.00 tff(not_type, type, (
% 4.30/3.00 not: $i > $i)).
% 4.30/3.00 tff(necessarily_type, type, (
% 4.30/3.00 necessarily: $i > $i)).
% 4.30/3.00 tff(implies_type, type, (
% 4.30/3.00 implies: ( $i * $i ) > $i)).
% 4.30/3.00 tff(op_strict_implies_type, type, (
% 4.30/3.00 op_strict_implies: $o)).
% 4.30/3.00 tff(strict_equiv_type, type, (
% 4.30/3.00 strict_equiv: ( $i * $i ) > $i)).
% 4.30/3.00 tff(op_strict_equiv_type, type, (
% 4.30/3.00 op_strict_equiv: $o)).
% 4.30/3.00 tff(axiom_m1_type, type, (
% 4.30/3.00 axiom_m1: $o)).
% 4.30/3.00 tff(adjunction_type, type, (
% 4.30/3.00 adjunction: $o)).
% 4.30/3.00 tff(substitution_strict_equiv_type, type, (
% 4.30/3.00 substitution_strict_equiv: $o)).
% 4.30/3.00 tff(axiom_m4_type, type, (
% 4.30/3.00 axiom_m4: $o)).
% 4.30/3.00 tff(axiom_m5_type, type, (
% 4.30/3.00 axiom_m5: $o)).
% 4.30/3.00 tff(op_implies_and_type, type, (
% 4.30/3.00 op_implies_and: $o)).
% 4.30/3.00 tff(axiom_m3_type, type, (
% 4.30/3.00 axiom_m3: $o)).
% 4.30/3.00 tff(modus_ponens_type, type, (
% 4.30/3.00 modus_ponens: $o)).
% 4.30/3.00 tff(modus_ponens_strict_implies_type, type, (
% 4.30/3.00 modus_ponens_strict_implies: $o)).
% 4.30/3.00 tff(1,plain,
% 4.30/3.00 (^[X: $i, Y: $i] : refl((strict_implies(X, Y) = necessarily(implies(X, Y))) <=> (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 4.30/3.00 inference(bind,[status(th)],[])).
% 4.30/3.00 tff(2,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(quant_intro,[status(thm)],[1])).
% 4.30/3.00 tff(3,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(4,plain,
% 4.30/3.00 (($false | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(5,plain,
% 4.30/3.00 ((~$true) <=> $false),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(6,axiom,(op_strict_implies), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_op_strict_implies')).
% 4.30/3.00 tff(7,plain,
% 4.30/3.00 (op_strict_implies <=> $true),
% 4.30/3.00 inference(iff_true,[status(thm)],[6])).
% 4.30/3.00 tff(8,plain,
% 4.30/3.00 ((~op_strict_implies) <=> (~$true)),
% 4.30/3.00 inference(monotonicity,[status(thm)],[7])).
% 4.30/3.00 tff(9,plain,
% 4.30/3.00 ((~op_strict_implies) <=> $false),
% 4.30/3.00 inference(transitivity,[status(thm)],[8, 5])).
% 4.30/3.00 tff(10,plain,
% 4.30/3.00 (((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ($false | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[9])).
% 4.30/3.00 tff(11,plain,
% 4.30/3.00 (((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(transitivity,[status(thm)],[10, 4])).
% 4.30/3.00 tff(12,plain,
% 4.30/3.00 (((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(13,plain,
% 4.30/3.00 ((op_strict_implies => ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(14,axiom,(op_strict_implies => ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+1.ax','op_strict_implies')).
% 4.30/3.00 tff(15,plain,
% 4.30/3.00 ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[14, 13])).
% 4.30/3.00 tff(16,plain,
% 4.30/3.00 ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[15, 12])).
% 4.30/3.00 tff(17,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[16, 11])).
% 4.30/3.00 tff(18,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[17, 3])).
% 4.30/3.00 tff(19,plain,(
% 4.30/3.00 ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(skolemize,[status(sab)],[18])).
% 4.30/3.00 tff(20,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[19, 2])).
% 4.30/3.00 tff(21,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)) = necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(22,plain,
% 4.30/3.00 (strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)) = necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[21, 20])).
% 4.30/3.00 tff(23,plain,
% 4.30/3.00 (necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))) = strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.00 inference(symmetry,[status(thm)],[22])).
% 4.30/3.00 tff(24,plain,
% 4.30/3.00 (and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))) = and(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[23, 23])).
% 4.30/3.00 tff(25,plain,
% 4.30/3.00 (and(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))) = and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.00 inference(symmetry,[status(thm)],[24])).
% 4.30/3.00 tff(26,plain,
% 4.30/3.00 (^[X: $i, Y: $i] : refl((strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))) <=> (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))))),
% 4.30/3.00 inference(bind,[status(th)],[])).
% 4.30/3.00 tff(27,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))) <=> ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(quant_intro,[status(thm)],[26])).
% 4.30/3.00 tff(28,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))) <=> ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(29,plain,
% 4.30/3.00 (($false | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) <=> ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(30,axiom,(op_strict_equiv), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_op_strict_equiv')).
% 4.30/3.00 tff(31,plain,
% 4.30/3.00 (op_strict_equiv <=> $true),
% 4.30/3.00 inference(iff_true,[status(thm)],[30])).
% 4.30/3.00 tff(32,plain,
% 4.30/3.00 ((~op_strict_equiv) <=> (~$true)),
% 4.30/3.00 inference(monotonicity,[status(thm)],[31])).
% 4.30/3.00 tff(33,plain,
% 4.30/3.00 ((~op_strict_equiv) <=> $false),
% 4.30/3.00 inference(transitivity,[status(thm)],[32, 5])).
% 4.30/3.00 tff(34,plain,
% 4.30/3.00 (((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) <=> ($false | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[33])).
% 4.30/3.00 tff(35,plain,
% 4.30/3.00 (((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) <=> ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(transitivity,[status(thm)],[34, 29])).
% 4.30/3.00 tff(36,plain,
% 4.30/3.00 (((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) <=> ((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(37,plain,
% 4.30/3.00 ((op_strict_equiv => ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) <=> ((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(38,axiom,(op_strict_equiv => ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+1.ax','op_strict_equiv')).
% 4.30/3.00 tff(39,plain,
% 4.30/3.00 ((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[38, 37])).
% 4.30/3.00 tff(40,plain,
% 4.30/3.00 ((~op_strict_equiv) | ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[39, 36])).
% 4.30/3.00 tff(41,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[40, 35])).
% 4.30/3.00 tff(42,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[41, 28])).
% 4.30/3.00 tff(43,plain,(
% 4.30/3.00 ![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(skolemize,[status(sab)],[42])).
% 4.30/3.00 tff(44,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[43, 27])).
% 4.30/3.00 tff(45,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(not(Y!3), X!4), and(not(Y!3), X!4)) = and(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(46,plain,
% 4.30/3.00 (strict_equiv(and(not(Y!3), X!4), and(not(Y!3), X!4)) = and(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[45, 44])).
% 4.30/3.00 tff(47,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(48,plain,
% 4.30/3.00 (strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[47, 44])).
% 4.30/3.00 tff(49,plain,
% 4.30/3.00 (and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))) = strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4))),
% 4.30/3.00 inference(symmetry,[status(thm)],[48])).
% 4.30/3.00 tff(50,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))) = necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(51,plain,
% 4.30/3.00 (strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))) = necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[50, 20])).
% 4.30/3.00 tff(52,plain,
% 4.30/3.00 (necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))) = strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))),
% 4.30/3.00 inference(symmetry,[status(thm)],[51])).
% 4.30/3.00 tff(53,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)) = necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(54,plain,
% 4.30/3.00 (strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)) = necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[53, 20])).
% 4.30/3.00 tff(55,plain,
% 4.30/3.00 (necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))) = strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))),
% 4.30/3.00 inference(symmetry,[status(thm)],[54])).
% 4.30/3.00 tff(56,plain,
% 4.30/3.00 (and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[55, 52])).
% 4.30/3.00 tff(57,plain,
% 4.30/3.00 (and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4))),
% 4.30/3.00 inference(transitivity,[status(thm)],[56, 49])).
% 4.30/3.00 tff(58,plain,
% 4.30/3.00 (is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) <=> is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[57])).
% 4.30/3.00 tff(59,plain,
% 4.30/3.00 (is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) <=> is_a_theorem(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[55])).
% 4.30/3.00 tff(60,plain,
% 4.30/3.00 (is_a_theorem(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))) <=> is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.00 inference(symmetry,[status(thm)],[59])).
% 4.30/3.00 tff(61,plain,
% 4.30/3.00 (^[X: $i, Y: $i] : refl(is_a_theorem(strict_implies(and(X, Y), and(Y, X))) <=> is_a_theorem(strict_implies(and(X, Y), and(Y, X))))),
% 4.30/3.00 inference(bind,[status(th)],[])).
% 4.30/3.00 tff(62,plain,
% 4.30/3.00 (![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X))) <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(quant_intro,[status(thm)],[61])).
% 4.30/3.00 tff(63,plain,
% 4.30/3.00 (![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X))) <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(64,plain,
% 4.30/3.00 (($true <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(65,axiom,(axiom_m1), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_axiom_m1')).
% 4.30/3.00 tff(66,plain,
% 4.30/3.00 (axiom_m1 <=> $true),
% 4.30/3.00 inference(iff_true,[status(thm)],[65])).
% 4.30/3.00 tff(67,plain,
% 4.30/3.00 ((axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) <=> ($true <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X))))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[66])).
% 4.30/3.00 tff(68,plain,
% 4.30/3.00 ((axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(transitivity,[status(thm)],[67, 64])).
% 4.30/3.00 tff(69,plain,
% 4.30/3.00 ((axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) <=> (axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(70,axiom,(axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','axiom_m1')).
% 4.30/3.00 tff(71,plain,
% 4.30/3.00 (axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[70, 69])).
% 4.30/3.00 tff(72,plain,
% 4.30/3.00 (axiom_m1 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[71, 69])).
% 4.30/3.00 tff(73,plain,
% 4.30/3.00 (![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[72, 68])).
% 4.30/3.00 tff(74,plain,
% 4.30/3.00 (![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[73, 63])).
% 4.30/3.00 tff(75,plain,(
% 4.30/3.00 ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(skolemize,[status(sab)],[74])).
% 4.30/3.00 tff(76,plain,
% 4.30/3.00 (![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[75, 62])).
% 4.30/3.00 tff(77,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(78,plain,
% 4.30/3.00 (is_a_theorem(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[77, 76])).
% 4.30/3.00 tff(79,plain,
% 4.30/3.00 (is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[78, 60])).
% 4.30/3.00 tff(80,plain,
% 4.30/3.00 (is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) <=> is_a_theorem(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[52])).
% 4.30/3.00 tff(81,plain,
% 4.30/3.00 (is_a_theorem(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))) <=> is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.00 inference(symmetry,[status(thm)],[80])).
% 4.30/3.00 tff(82,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(83,plain,
% 4.30/3.00 (is_a_theorem(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.00 inference(unit_resolution,[status(thm)],[82, 76])).
% 4.30/3.00 tff(84,plain,
% 4.30/3.00 (is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[83, 81])).
% 4.30/3.00 tff(85,plain,
% 4.30/3.00 (^[X: $i, Y: $i] : refl((is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X))) <=> (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X))))),
% 4.30/3.00 inference(bind,[status(th)],[])).
% 4.30/3.00 tff(86,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X))) <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))),
% 4.30/3.00 inference(quant_intro,[status(thm)],[85])).
% 4.30/3.00 tff(87,plain,
% 4.30/3.00 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((is_a_theorem(X) & is_a_theorem(Y)) <=> (~((~is_a_theorem(Y)) | (~is_a_theorem(X))))), ((~(is_a_theorem(X) & is_a_theorem(Y))) <=> (~(~((~is_a_theorem(Y)) | (~is_a_theorem(X))))))), rewrite((~(~((~is_a_theorem(Y)) | (~is_a_theorem(X))))) <=> ((~is_a_theorem(Y)) | (~is_a_theorem(X)))), ((~(is_a_theorem(X) & is_a_theorem(Y))) <=> ((~is_a_theorem(Y)) | (~is_a_theorem(X))))), ((is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))) <=> (is_a_theorem(and(X, Y)) | ((~is_a_theorem(Y)) | (~is_a_theorem(X)))))), rewrite((is_a_theorem(and(X, Y)) | ((~is_a_theorem(Y)) | (~is_a_theorem(X)))) <=> (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))), ((is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))) <=> (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))))),
% 4.30/3.00 inference(bind,[status(th)],[])).
% 4.30/3.00 tff(88,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))) <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))),
% 4.30/3.00 inference(quant_intro,[status(thm)],[87])).
% 4.30/3.00 tff(89,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))) <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(90,plain,
% 4.30/3.00 (($true <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))) <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(91,axiom,(adjunction), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_adjunction')).
% 4.30/3.00 tff(92,plain,
% 4.30/3.00 (adjunction <=> $true),
% 4.30/3.00 inference(iff_true,[status(thm)],[91])).
% 4.30/3.00 tff(93,plain,
% 4.30/3.00 ((adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))) <=> ($true <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))))),
% 4.30/3.00 inference(monotonicity,[status(thm)],[92])).
% 4.30/3.00 tff(94,plain,
% 4.30/3.00 ((adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))) <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(transitivity,[status(thm)],[93, 90])).
% 4.30/3.00 tff(95,plain,
% 4.30/3.00 ((adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))) <=> (adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(96,plain,
% 4.30/3.00 ((adjunction <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(Y)) => is_a_theorem(and(X, Y)))) <=> (adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y)))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(97,axiom,(adjunction <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(Y)) => is_a_theorem(and(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','adjunction')).
% 4.30/3.00 tff(98,plain,
% 4.30/3.00 (adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[97, 96])).
% 4.30/3.00 tff(99,plain,
% 4.30/3.00 (adjunction <=> ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[98, 95])).
% 4.30/3.00 tff(100,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[99, 94])).
% 4.30/3.00 tff(101,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[100, 89])).
% 4.30/3.00 tff(102,plain,(
% 4.30/3.00 ![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~(is_a_theorem(X) & is_a_theorem(Y))))),
% 4.30/3.00 inference(skolemize,[status(sab)],[101])).
% 4.30/3.00 tff(103,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[102, 88])).
% 4.30/3.00 tff(104,plain,
% 4.30/3.00 (![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[103, 86])).
% 4.30/3.00 tff(105,plain,
% 4.30/3.00 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))))),
% 4.30/3.00 inference(rewrite,[status(thm)],[])).
% 4.30/3.00 tff(106,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))))),
% 4.30/3.00 inference(quant_inst,[status(thm)],[])).
% 4.30/3.00 tff(107,plain,
% 4.30/3.00 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))),
% 4.30/3.00 inference(modus_ponens,[status(thm)],[106, 105])).
% 4.30/3.00 tff(108,plain,
% 4.30/3.00 (is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[107, 104])).
% 4.30/3.01 tff(109,plain,
% 4.30/3.01 (is_a_theorem(and(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[108, 84, 79])).
% 4.30/3.01 tff(110,plain,
% 4.30/3.01 (is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[109, 58])).
% 4.30/3.01 tff(111,plain,
% 4.30/3.01 (^[X: $i, Y: $i] : refl(((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)) <=> ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)))),
% 4.30/3.01 inference(bind,[status(th)],[])).
% 4.30/3.01 tff(112,plain,
% 4.30/3.01 (![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)) <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(quant_intro,[status(thm)],[111])).
% 4.30/3.01 tff(113,plain,
% 4.30/3.01 (![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)) <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(rewrite,[status(thm)],[])).
% 4.30/3.01 tff(114,plain,
% 4.30/3.01 (($true <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(rewrite,[status(thm)],[])).
% 4.30/3.01 tff(115,axiom,(substitution_strict_equiv), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_substitution_strict_equiv')).
% 4.30/3.01 tff(116,plain,
% 4.30/3.01 (substitution_strict_equiv <=> $true),
% 4.30/3.01 inference(iff_true,[status(thm)],[115])).
% 4.30/3.01 tff(117,plain,
% 4.30/3.01 ((substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) <=> ($true <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[116])).
% 4.30/3.01 tff(118,plain,
% 4.30/3.01 ((substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(transitivity,[status(thm)],[117, 114])).
% 4.30/3.01 tff(119,plain,
% 4.30/3.01 ((substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) <=> (substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)))),
% 4.30/3.01 inference(rewrite,[status(thm)],[])).
% 4.30/3.01 tff(120,plain,
% 4.30/3.01 ((substitution_strict_equiv <=> ![X: $i, Y: $i] : (is_a_theorem(strict_equiv(X, Y)) => (X = Y))) <=> (substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y)))),
% 4.30/3.01 inference(rewrite,[status(thm)],[])).
% 4.30/3.01 tff(121,axiom,(substitution_strict_equiv <=> ![X: $i, Y: $i] : (is_a_theorem(strict_equiv(X, Y)) => (X = Y))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','substitution_strict_equiv')).
% 4.30/3.01 tff(122,plain,
% 4.30/3.01 (substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[121, 120])).
% 4.30/3.01 tff(123,plain,
% 4.30/3.01 (substitution_strict_equiv <=> ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[122, 119])).
% 4.30/3.01 tff(124,plain,
% 4.30/3.01 (![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[123, 118])).
% 4.30/3.01 tff(125,plain,
% 4.30/3.01 (![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[124, 113])).
% 4.30/3.01 tff(126,plain,(
% 4.30/3.01 ![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(skolemize,[status(sab)],[125])).
% 4.30/3.01 tff(127,plain,
% 4.30/3.01 (![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[126, 112])).
% 4.30/3.01 tff(128,plain,
% 4.30/3.01 (((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) | (and(X!4, not(Y!3)) = and(not(Y!3), X!4)))) <=> ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) | (and(X!4, not(Y!3)) = and(not(Y!3), X!4)))),
% 4.30/3.01 inference(rewrite,[status(thm)],[])).
% 4.30/3.01 tff(129,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) | (and(X!4, not(Y!3)) = and(not(Y!3), X!4)))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(130,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) | (and(X!4, not(Y!3)) = and(not(Y!3), X!4))),
% 4.30/3.01 inference(modus_ponens,[status(thm)],[129, 128])).
% 4.30/3.01 tff(131,plain,
% 4.30/3.01 ((~is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) | (and(X!4, not(Y!3)) = and(not(Y!3), X!4))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[130, 127])).
% 4.30/3.01 tff(132,plain,
% 4.30/3.01 (and(X!4, not(Y!3)) = and(not(Y!3), X!4)),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[131, 110])).
% 4.30/3.01 tff(133,plain,
% 4.30/3.01 (strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)) = strict_equiv(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[132])).
% 4.30/3.01 tff(134,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(135,plain,
% 4.30/3.01 (strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[134, 20])).
% 4.30/3.01 tff(136,plain,
% 4.30/3.01 (necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))) = strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[135])).
% 4.30/3.01 tff(137,plain,
% 4.30/3.01 (and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))) = and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[136])).
% 4.30/3.01 tff(138,plain,
% 4.30/3.01 (and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))) = and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[137])).
% 4.30/3.01 tff(139,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(140,plain,
% 4.30/3.01 (strict_equiv(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[139, 44])).
% 4.30/3.01 tff(141,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))) = necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(142,plain,
% 4.30/3.01 (strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))) = necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[141, 20])).
% 4.30/3.01 tff(143,plain,
% 4.30/3.01 (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))),
% 4.30/3.01 inference(symmetry,[status(thm)],[142])).
% 4.30/3.01 tff(144,plain,
% 4.30/3.01 (strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = strict_equiv(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[143])).
% 4.30/3.01 tff(145,plain,
% 4.30/3.01 (strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))))),
% 4.30/3.01 inference(transitivity,[status(thm)],[144, 140, 138])).
% 4.30/3.01 tff(146,plain,
% 4.30/3.01 (is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) <=> is_a_theorem(and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[145])).
% 4.30/3.01 tff(147,plain,
% 4.30/3.01 (is_a_theorem(and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))))) <=> is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[146])).
% 4.30/3.01 tff(148,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))) = necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(149,plain,
% 4.30/3.01 (strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))) = necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[148, 20])).
% 4.30/3.01 tff(150,plain,
% 4.30/3.01 (necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))) = strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.01 inference(symmetry,[status(thm)],[149])).
% 4.30/3.01 tff(151,plain,
% 4.30/3.01 (and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))) = and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[150])).
% 4.30/3.01 tff(152,plain,
% 4.30/3.01 (and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))) = and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[151])).
% 4.30/3.01 tff(153,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(154,plain,
% 4.30/3.01 (strict_equiv(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[153, 44])).
% 4.30/3.01 tff(155,plain,
% 4.30/3.01 (strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))) = strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[132])).
% 4.30/3.01 tff(156,plain,
% 4.30/3.01 (strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)) = strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))),
% 4.30/3.01 inference(symmetry,[status(thm)],[155])).
% 4.30/3.01 tff(157,plain,
% 4.30/3.01 (strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)) = necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.01 inference(transitivity,[status(thm)],[156, 51])).
% 4.30/3.01 tff(158,plain,
% 4.30/3.01 (strict_equiv(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = strict_equiv(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[157])).
% 4.30/3.01 tff(159,plain,
% 4.30/3.01 (strict_equiv(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = strict_equiv(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[158])).
% 4.30/3.01 tff(160,plain,
% 4.30/3.01 (strict_equiv(and(not(Y!3), X!4), and(not(Y!3), X!4)) = strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4))),
% 4.30/3.01 inference(symmetry,[status(thm)],[133])).
% 4.30/3.01 tff(161,plain,
% 4.30/3.01 (strict_equiv(and(not(Y!3), X!4), and(X!4, not(Y!3))) = strict_equiv(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[132])).
% 4.30/3.01 tff(162,plain,
% 4.30/3.01 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(not(Y!3), X!4), and(X!4, not(Y!3))) = and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(quant_inst,[status(thm)],[])).
% 4.30/3.01 tff(163,plain,
% 4.30/3.01 (strict_equiv(and(not(Y!3), X!4), and(X!4, not(Y!3))) = and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.01 inference(unit_resolution,[status(thm)],[162, 44])).
% 4.30/3.01 tff(164,plain,
% 4.30/3.01 (and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))) = strict_equiv(and(not(Y!3), X!4), and(X!4, not(Y!3)))),
% 4.30/3.01 inference(symmetry,[status(thm)],[163])).
% 4.30/3.01 tff(165,plain,
% 4.30/3.01 (and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.01 inference(transitivity,[status(thm)],[164, 161, 160, 48])).
% 4.30/3.01 tff(166,plain,
% 4.30/3.01 (strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)) = strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[132])).
% 4.30/3.01 tff(167,plain,
% 4.30/3.01 (strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)) = strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))),
% 4.30/3.01 inference(symmetry,[status(thm)],[166])).
% 4.30/3.01 tff(168,plain,
% 4.30/3.01 (necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))) = necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.01 inference(transitivity,[status(thm)],[52, 155, 167, 54])).
% 4.30/3.01 tff(169,plain,
% 4.30/3.01 (strict_equiv(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[168, 165])).
% 4.30/3.01 tff(170,plain,
% 4.30/3.01 (strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = strict_equiv(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[169])).
% 4.30/3.01 tff(171,plain,
% 4.30/3.01 (and(not(Y!3), X!4) = and(X!4, not(Y!3))),
% 4.30/3.01 inference(symmetry,[status(thm)],[132])).
% 4.30/3.01 tff(172,plain,
% 4.30/3.01 (strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)) = strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[171, 171])).
% 4.30/3.01 tff(173,plain,
% 4.30/3.01 (strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))) = strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.01 inference(symmetry,[status(thm)],[172])).
% 4.30/3.01 tff(174,plain,
% 4.30/3.01 (strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))) = necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.01 inference(transitivity,[status(thm)],[173, 167, 54])).
% 4.30/3.01 tff(175,plain,
% 4.30/3.01 (strict_equiv(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(monotonicity,[status(thm)],[174])).
% 4.30/3.01 tff(176,plain,
% 4.30/3.01 (and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), strict_implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))) = strict_equiv(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.01 inference(symmetry,[status(thm)],[140])).
% 4.30/3.01 tff(177,plain,
% 4.30/3.01 (and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))) = and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.01 inference(transitivity,[status(thm)],[137, 176, 175, 170, 159, 154, 152])).
% 4.30/3.01 tff(178,plain,
% 4.30/3.01 (is_a_theorem(and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))))))) <=> is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[177])).
% 4.30/3.02 tff(179,plain,
% 4.30/3.02 (is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) <=> is_a_theorem(and(strict_implies(strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))), necessarily(implies(and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), strict_implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))))))),
% 4.30/3.02 inference(symmetry,[status(thm)],[178])).
% 4.30/3.02 tff(180,plain,
% 4.30/3.02 (is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) <=> is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.02 inference(transitivity,[status(thm)],[179, 147])).
% 4.30/3.02 tff(181,plain,
% 4.30/3.02 (strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = strict_implies(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[157])).
% 4.30/3.02 tff(182,plain,
% 4.30/3.02 (strict_implies(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(symmetry,[status(thm)],[181])).
% 4.30/3.02 tff(183,plain,
% 4.30/3.02 (and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))) = and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(transitivity,[status(thm)],[164, 161, 46, 25])).
% 4.30/3.02 tff(184,plain,
% 4.30/3.02 (necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))) = necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.02 inference(transitivity,[status(thm)],[52, 155, 22])).
% 4.30/3.02 tff(185,plain,
% 4.30/3.02 (strict_implies(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))) = strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[184, 183])).
% 4.30/3.02 tff(186,plain,
% 4.30/3.02 (strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))) = strict_implies(necessarily(implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(symmetry,[status(thm)],[185])).
% 4.30/3.02 tff(187,plain,
% 4.30/3.02 (and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))) = and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(transitivity,[status(thm)],[49, 133, 46, 25])).
% 4.30/3.02 tff(188,plain,
% 4.30/3.02 (necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))) = necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.02 inference(transitivity,[status(thm)],[55, 166, 22])).
% 4.30/3.02 tff(189,plain,
% 4.30/3.02 (strict_implies(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[188, 187])).
% 4.30/3.02 tff(190,plain,
% 4.30/3.02 (strict_implies(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))) = strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(transitivity,[status(thm)],[189, 186, 182])).
% 4.30/3.02 tff(191,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))) <=> is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[190])).
% 4.30/3.02 tff(192,plain,
% 4.30/3.02 (strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))) = strict_implies(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.02 inference(symmetry,[status(thm)],[189])).
% 4.30/3.02 tff(193,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) <=> is_a_theorem(strict_implies(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[192])).
% 4.30/3.02 tff(194,plain,
% 4.30/3.02 (^[X: $i] : refl(is_a_theorem(strict_implies(X, and(X, X))) <=> is_a_theorem(strict_implies(X, and(X, X))))),
% 4.30/3.02 inference(bind,[status(th)],[])).
% 4.30/3.02 tff(195,plain,
% 4.30/3.02 (![X: $i] : is_a_theorem(strict_implies(X, and(X, X))) <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(quant_intro,[status(thm)],[194])).
% 4.30/3.02 tff(196,plain,
% 4.30/3.02 (![X: $i] : is_a_theorem(strict_implies(X, and(X, X))) <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(197,plain,
% 4.30/3.02 (($true <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))) <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(198,axiom,(axiom_m4), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_axiom_m4')).
% 4.30/3.02 tff(199,plain,
% 4.30/3.02 (axiom_m4 <=> $true),
% 4.30/3.02 inference(iff_true,[status(thm)],[198])).
% 4.30/3.02 tff(200,plain,
% 4.30/3.02 ((axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))) <=> ($true <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[199])).
% 4.30/3.02 tff(201,plain,
% 4.30/3.02 ((axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))) <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(transitivity,[status(thm)],[200, 197])).
% 4.30/3.02 tff(202,plain,
% 4.30/3.02 ((axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))) <=> (axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X))))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(203,axiom,(axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','axiom_m4')).
% 4.30/3.02 tff(204,plain,
% 4.30/3.02 (axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[203, 202])).
% 4.30/3.02 tff(205,plain,
% 4.30/3.02 (axiom_m4 <=> ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[204, 202])).
% 4.30/3.02 tff(206,plain,
% 4.30/3.02 (![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[205, 201])).
% 4.30/3.02 tff(207,plain,
% 4.30/3.02 (![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[206, 196])).
% 4.30/3.02 tff(208,plain,(
% 4.30/3.02 ![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(skolemize,[status(sab)],[207])).
% 4.30/3.02 tff(209,plain,
% 4.30/3.02 (![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[208, 195])).
% 4.30/3.02 tff(210,plain,
% 4.30/3.02 ((~![X: $i] : is_a_theorem(strict_implies(X, and(X, X)))) | is_a_theorem(strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))))),
% 4.30/3.02 inference(quant_inst,[status(thm)],[])).
% 4.30/3.02 tff(211,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))))),
% 4.30/3.02 inference(unit_resolution,[status(thm)],[210, 209])).
% 4.30/3.02 tff(212,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(necessarily(implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[211, 193])).
% 4.30/3.02 tff(213,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[212, 191])).
% 4.30/3.02 tff(214,plain,
% 4.30/3.02 (is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))) <=> is_a_theorem(strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[150])).
% 4.30/3.02 tff(215,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))) <=> is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.02 inference(symmetry,[status(thm)],[214])).
% 4.30/3.02 tff(216,plain,
% 4.30/3.02 (^[X: $i, Y: $i, Z: $i] : refl(is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z))) <=> is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z))))),
% 4.30/3.02 inference(bind,[status(th)],[])).
% 4.30/3.02 tff(217,plain,
% 4.30/3.02 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(quant_intro,[status(thm)],[216])).
% 4.30/3.02 tff(218,plain,
% 4.30/3.02 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(219,plain,
% 4.30/3.02 (($true <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(220,axiom,(axiom_m5), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_axiom_m5')).
% 4.30/3.02 tff(221,plain,
% 4.30/3.02 (axiom_m5 <=> $true),
% 4.30/3.02 inference(iff_true,[status(thm)],[220])).
% 4.30/3.02 tff(222,plain,
% 4.30/3.02 ((axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))) <=> ($true <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z))))),
% 4.30/3.02 inference(monotonicity,[status(thm)],[221])).
% 4.30/3.02 tff(223,plain,
% 4.30/3.02 ((axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(transitivity,[status(thm)],[222, 219])).
% 4.30/3.02 tff(224,plain,
% 4.30/3.02 ((axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))) <=> (axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z))))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(225,axiom,(axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','axiom_m5')).
% 4.30/3.02 tff(226,plain,
% 4.30/3.02 (axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[225, 224])).
% 4.30/3.02 tff(227,plain,
% 4.30/3.02 (axiom_m5 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[226, 224])).
% 4.30/3.02 tff(228,plain,
% 4.30/3.02 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[227, 223])).
% 4.30/3.02 tff(229,plain,
% 4.30/3.02 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[228, 218])).
% 4.30/3.02 tff(230,plain,(
% 4.30/3.02 ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(skolemize,[status(sab)],[229])).
% 4.30/3.02 tff(231,plain,
% 4.30/3.02 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[230, 217])).
% 4.30/3.02 tff(232,plain,
% 4.30/3.02 ((~![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(strict_implies(X, Y), strict_implies(Y, Z)), strict_implies(X, Z)))) | is_a_theorem(strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(quant_inst,[status(thm)],[])).
% 4.30/3.02 tff(233,plain,
% 4.30/3.02 (is_a_theorem(strict_implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.02 inference(unit_resolution,[status(thm)],[232, 231])).
% 4.30/3.02 tff(234,plain,
% 4.30/3.02 (is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[233, 215])).
% 4.30/3.02 tff(235,plain,
% 4.30/3.02 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(236,plain,
% 4.30/3.02 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))))))),
% 4.30/3.02 inference(quant_inst,[status(thm)],[])).
% 4.30/3.02 tff(237,plain,
% 4.30/3.02 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[236, 235])).
% 4.30/3.02 tff(238,plain,
% 4.30/3.02 (is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))) | (~is_a_theorem(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))))))),
% 4.30/3.02 inference(unit_resolution,[status(thm)],[237, 104])).
% 4.30/3.02 tff(239,plain,
% 4.30/3.02 (is_a_theorem(and(strict_implies(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)))), necessarily(implies(and(strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))), strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4))), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))))),
% 4.30/3.02 inference(unit_resolution,[status(thm)],[238, 234, 213])).
% 4.30/3.02 tff(240,plain,
% 4.30/3.02 (is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.02 inference(modus_ponens,[status(thm)],[239, 180])).
% 4.30/3.02 tff(241,plain,
% 4.30/3.02 (((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))) | (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))) <=> ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))) | (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.02 inference(rewrite,[status(thm)],[])).
% 4.30/3.02 tff(242,plain,
% 4.30/3.02 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))) | (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))),
% 4.30/3.02 inference(quant_inst,[status(thm)],[])).
% 4.30/3.02 tff(243,plain,
% 4.30/3.02 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))) | (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[242, 241])).
% 4.30/3.03 tff(244,plain,
% 4.30/3.03 ((~is_a_theorem(strict_equiv(necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))), and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))))) | (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[243, 127])).
% 4.30/3.03 tff(245,plain,
% 4.30/3.03 (necessarily(implies(and(X!4, not(Y!3)), and(X!4, not(Y!3)))) = and(strict_implies(and(X!4, not(Y!3)), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(X!4, not(Y!3))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[244, 240])).
% 4.30/3.03 tff(246,plain,
% 4.30/3.03 (implies(and(not(Y!3), X!4), and(X!4, not(Y!3))) = implies(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[132])).
% 4.30/3.03 tff(247,plain,
% 4.30/3.03 (^[X: $i, Y: $i] : refl((implies(X, Y) = not(and(X, not(Y)))) <=> (implies(X, Y) = not(and(X, not(Y)))))),
% 4.30/3.03 inference(bind,[status(th)],[])).
% 4.30/3.03 tff(248,plain,
% 4.30/3.03 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(quant_intro,[status(thm)],[247])).
% 4.30/3.03 tff(249,plain,
% 4.30/3.03 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(250,plain,
% 4.30/3.03 (($false | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(251,axiom,(op_implies_and), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','hilbert_op_implies_and')).
% 4.30/3.03 tff(252,plain,
% 4.30/3.03 (op_implies_and <=> $true),
% 4.30/3.03 inference(iff_true,[status(thm)],[251])).
% 4.30/3.03 tff(253,plain,
% 4.30/3.03 ((~op_implies_and) <=> (~$true)),
% 4.30/3.03 inference(monotonicity,[status(thm)],[252])).
% 4.30/3.03 tff(254,plain,
% 4.30/3.03 ((~op_implies_and) <=> $false),
% 4.30/3.03 inference(transitivity,[status(thm)],[253, 5])).
% 4.30/3.03 tff(255,plain,
% 4.30/3.03 (((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ($false | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[254])).
% 4.30/3.03 tff(256,plain,
% 4.30/3.03 (((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(transitivity,[status(thm)],[255, 250])).
% 4.30/3.03 tff(257,plain,
% 4.30/3.03 (((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(258,plain,
% 4.30/3.03 ((op_implies_and => ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(259,axiom,(op_implies_and => ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+1.ax','op_implies_and')).
% 4.30/3.03 tff(260,plain,
% 4.30/3.03 ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[259, 258])).
% 4.30/3.03 tff(261,plain,
% 4.30/3.03 ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[260, 257])).
% 4.30/3.03 tff(262,plain,
% 4.30/3.03 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[261, 256])).
% 4.30/3.03 tff(263,plain,
% 4.30/3.03 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[262, 249])).
% 4.30/3.03 tff(264,plain,(
% 4.30/3.03 ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(skolemize,[status(sab)],[263])).
% 4.30/3.03 tff(265,plain,
% 4.30/3.03 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[264, 248])).
% 4.30/3.03 tff(266,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) | (implies(and(not(Y!3), X!4), and(X!4, not(Y!3))) = not(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(267,plain,
% 4.30/3.03 (implies(and(not(Y!3), X!4), and(X!4, not(Y!3))) = not(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[266, 265])).
% 4.30/3.03 tff(268,plain,
% 4.30/3.03 (not(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))) = implies(and(not(Y!3), X!4), and(X!4, not(Y!3)))),
% 4.30/3.03 inference(symmetry,[status(thm)],[267])).
% 4.30/3.03 tff(269,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))) = and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(270,plain,
% 4.30/3.03 (strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))) = and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[269, 44])).
% 4.30/3.03 tff(271,plain,
% 4.30/3.03 (and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) = strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(symmetry,[status(thm)],[270])).
% 4.30/3.03 tff(272,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))) = necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(273,plain,
% 4.30/3.03 (strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))) = necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[272, 20])).
% 4.30/3.03 tff(274,plain,
% 4.30/3.03 (necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))) = strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(symmetry,[status(thm)],[273])).
% 4.30/3.03 tff(275,plain,
% 4.30/3.03 (and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) = and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[274])).
% 4.30/3.03 tff(276,plain,
% 4.30/3.03 (and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) = strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(transitivity,[status(thm)],[275, 271])).
% 4.30/3.03 tff(277,plain,
% 4.30/3.03 (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) <=> is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[276])).
% 4.30/3.03 tff(278,plain,
% 4.30/3.03 (is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) <=> is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[274])).
% 4.30/3.03 tff(279,plain,
% 4.30/3.03 (is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))) <=> is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.03 inference(symmetry,[status(thm)],[278])).
% 4.30/3.03 tff(280,plain,
% 4.30/3.03 (^[X: $i, Y: $i, Z: $i] : refl(is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z)))) <=> is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z)))))),
% 4.30/3.03 inference(bind,[status(th)],[])).
% 4.30/3.03 tff(281,plain,
% 4.30/3.03 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z)))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(quant_intro,[status(thm)],[280])).
% 4.30/3.03 tff(282,plain,
% 4.30/3.03 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z)))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(283,plain,
% 4.30/3.03 (($true <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(284,axiom,(axiom_m3), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_axiom_m3')).
% 4.30/3.03 tff(285,plain,
% 4.30/3.03 (axiom_m3 <=> $true),
% 4.30/3.03 inference(iff_true,[status(thm)],[284])).
% 4.30/3.03 tff(286,plain,
% 4.30/3.03 ((axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))) <=> ($true <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z)))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[285])).
% 4.30/3.03 tff(287,plain,
% 4.30/3.03 ((axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))) <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(transitivity,[status(thm)],[286, 283])).
% 4.30/3.03 tff(288,plain,
% 4.30/3.03 ((axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))) <=> (axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z)))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(289,axiom,(axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','axiom_m3')).
% 4.30/3.03 tff(290,plain,
% 4.30/3.03 (axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[289, 288])).
% 4.30/3.03 tff(291,plain,
% 4.30/3.03 (axiom_m3 <=> ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[290, 288])).
% 4.30/3.03 tff(292,plain,
% 4.30/3.03 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[291, 287])).
% 4.30/3.03 tff(293,plain,
% 4.30/3.03 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[292, 282])).
% 4.30/3.03 tff(294,plain,(
% 4.30/3.03 ![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(skolemize,[status(sab)],[293])).
% 4.30/3.03 tff(295,plain,
% 4.30/3.03 (![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[294, 281])).
% 4.30/3.03 tff(296,plain,
% 4.30/3.03 ((~![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))) | is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(297,plain,
% 4.30/3.03 (is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[296, 295])).
% 4.30/3.03 tff(298,plain,
% 4.30/3.03 (is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[297, 279])).
% 4.30/3.03 tff(299,plain,
% 4.30/3.03 (not(and(not(Y!3), X!4)) = not(and(X!4, not(Y!3)))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[171])).
% 4.30/3.03 tff(300,plain,
% 4.30/3.03 (and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[299, 132])).
% 4.30/3.03 tff(301,plain,
% 4.30/3.03 (and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)) = and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3)))),
% 4.30/3.03 inference(symmetry,[status(thm)],[300])).
% 4.30/3.03 tff(302,plain,
% 4.30/3.03 (strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))) = strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[301])).
% 4.30/3.03 tff(303,plain,
% 4.30/3.03 (strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3)))) = strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.03 inference(symmetry,[status(thm)],[302])).
% 4.30/3.03 tff(304,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))) = and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))), strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(305,plain,
% 4.30/3.03 (strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))) = and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))), strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[304, 44])).
% 4.30/3.03 tff(306,plain,
% 4.30/3.03 (and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))), strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) = strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.03 inference(symmetry,[status(thm)],[305])).
% 4.30/3.03 tff(307,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))) = necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(308,plain,
% 4.30/3.03 (strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))) = necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[307, 20])).
% 4.30/3.03 tff(309,plain,
% 4.30/3.03 (necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) = strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))),
% 4.30/3.03 inference(symmetry,[status(thm)],[308])).
% 4.30/3.03 tff(310,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))) = necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(311,plain,
% 4.30/3.03 (strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))) = necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[310, 20])).
% 4.30/3.03 tff(312,plain,
% 4.30/3.03 (necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))) = strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.03 inference(symmetry,[status(thm)],[311])).
% 4.30/3.03 tff(313,plain,
% 4.30/3.03 (and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) = and(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))), strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[312, 309])).
% 4.30/3.03 tff(314,plain,
% 4.30/3.03 (and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) = strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.03 inference(transitivity,[status(thm)],[313, 306])).
% 4.30/3.03 tff(315,plain,
% 4.30/3.03 (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) <=> is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[314])).
% 4.30/3.03 tff(316,plain,
% 4.30/3.03 (is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) <=> is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[312])).
% 4.30/3.03 tff(317,plain,
% 4.30/3.03 (is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))) <=> is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))))),
% 4.30/3.03 inference(symmetry,[status(thm)],[316])).
% 4.30/3.03 tff(318,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(319,plain,
% 4.30/3.03 (is_a_theorem(strict_implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[318, 76])).
% 4.30/3.03 tff(320,plain,
% 4.30/3.03 (is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[319, 317])).
% 4.30/3.03 tff(321,plain,
% 4.30/3.03 (is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) <=> is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(monotonicity,[status(thm)],[309])).
% 4.30/3.03 tff(322,plain,
% 4.30/3.03 (is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) <=> is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(symmetry,[status(thm)],[321])).
% 4.30/3.03 tff(323,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(324,plain,
% 4.30/3.03 (is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.03 inference(unit_resolution,[status(thm)],[323, 76])).
% 4.30/3.03 tff(325,plain,
% 4.30/3.03 (is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.03 inference(modus_ponens,[status(thm)],[324, 322])).
% 4.30/3.03 tff(326,plain,
% 4.30/3.03 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))))))),
% 4.30/3.03 inference(rewrite,[status(thm)],[])).
% 4.30/3.03 tff(327,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))))))),
% 4.30/3.03 inference(quant_inst,[status(thm)],[])).
% 4.30/3.03 tff(328,plain,
% 4.30/3.03 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[327, 326])).
% 4.30/3.04 tff(329,plain,
% 4.30/3.04 (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[328, 104])).
% 4.30/3.04 tff(330,plain,
% 4.30/3.04 (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))), necessarily(implies(and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[329, 325, 320])).
% 4.30/3.04 tff(331,plain,
% 4.30/3.04 (is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[330, 315])).
% 4.30/3.04 tff(332,plain,
% 4.30/3.04 (((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) <=> ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.04 inference(rewrite,[status(thm)],[])).
% 4.30/3.04 tff(333,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(334,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[333, 332])).
% 4.30/3.04 tff(335,plain,
% 4.30/3.04 ((~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[334, 127])).
% 4.30/3.04 tff(336,plain,
% 4.30/3.04 (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[335, 331])).
% 4.30/3.04 tff(337,plain,
% 4.30/3.04 (and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)) = and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))),
% 4.30/3.04 inference(symmetry,[status(thm)],[336])).
% 4.30/3.04 tff(338,plain,
% 4.30/3.04 (and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3))) = and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))),
% 4.30/3.04 inference(transitivity,[status(thm)],[300, 337])).
% 4.30/3.04 tff(339,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))) = and(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))), strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(340,plain,
% 4.30/3.04 (strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))) = and(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))), strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[339, 44])).
% 4.30/3.04 tff(341,plain,
% 4.30/3.04 (and(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))), strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))) = strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))),
% 4.30/3.04 inference(symmetry,[status(thm)],[340])).
% 4.30/3.04 tff(342,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))) = necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(343,plain,
% 4.30/3.04 (strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))) = necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[342, 20])).
% 4.30/3.04 tff(344,plain,
% 4.30/3.04 (necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))) = strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.04 inference(symmetry,[status(thm)],[343])).
% 4.30/3.04 tff(345,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))) = necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(346,plain,
% 4.30/3.04 (strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))) = necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[345, 20])).
% 4.30/3.04 tff(347,plain,
% 4.30/3.04 (necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))) = strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))),
% 4.30/3.04 inference(symmetry,[status(thm)],[346])).
% 4.30/3.04 tff(348,plain,
% 4.30/3.04 (and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) = and(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))), strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(monotonicity,[status(thm)],[347, 344])).
% 4.30/3.04 tff(349,plain,
% 4.30/3.04 (and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) = strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))),
% 4.30/3.04 inference(transitivity,[status(thm)],[348, 341])).
% 4.30/3.04 tff(350,plain,
% 4.30/3.04 (is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) <=> is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(monotonicity,[status(thm)],[349])).
% 4.30/3.04 tff(351,plain,
% 4.30/3.04 (is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) <=> is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(monotonicity,[status(thm)],[347])).
% 4.30/3.04 tff(352,plain,
% 4.30/3.04 (is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))) <=> is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))))),
% 4.30/3.04 inference(symmetry,[status(thm)],[351])).
% 4.30/3.04 tff(353,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(354,plain,
% 4.30/3.04 (is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[353, 76])).
% 4.30/3.04 tff(355,plain,
% 4.30/3.04 (is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[354, 352])).
% 4.30/3.04 tff(356,plain,
% 4.30/3.04 (is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) <=> is_a_theorem(strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(monotonicity,[status(thm)],[344])).
% 4.30/3.04 tff(357,plain,
% 4.30/3.04 (is_a_theorem(strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))) <=> is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.04 inference(symmetry,[status(thm)],[356])).
% 4.30/3.04 tff(358,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(359,plain,
% 4.30/3.04 (is_a_theorem(strict_implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[358, 76])).
% 4.30/3.04 tff(360,plain,
% 4.30/3.04 (is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[359, 357])).
% 4.30/3.04 tff(361,plain,
% 4.30/3.04 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))))))),
% 4.30/3.04 inference(rewrite,[status(thm)],[])).
% 4.30/3.04 tff(362,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(363,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[362, 361])).
% 4.30/3.04 tff(364,plain,
% 4.30/3.04 (is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[363, 104])).
% 4.30/3.04 tff(365,plain,
% 4.30/3.04 (is_a_theorem(and(necessarily(implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))), necessarily(implies(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[364, 360, 355])).
% 4.30/3.04 tff(366,plain,
% 4.30/3.04 (is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[365, 350])).
% 4.30/3.04 tff(367,plain,
% 4.30/3.04 (((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) | (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) <=> ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) | (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(rewrite,[status(thm)],[])).
% 4.30/3.04 tff(368,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) | (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(369,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) | (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))),
% 4.30/3.04 inference(modus_ponens,[status(thm)],[368, 367])).
% 4.30/3.04 tff(370,plain,
% 4.30/3.04 ((~is_a_theorem(strict_equiv(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))) | (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[369, 127])).
% 4.30/3.04 tff(371,plain,
% 4.30/3.04 (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[370, 366])).
% 4.30/3.04 tff(372,plain,
% 4.30/3.04 (and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))),
% 4.30/3.04 inference(symmetry,[status(thm)],[371])).
% 4.30/3.04 tff(373,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (strict_equiv(X, Y) = and(strict_implies(X, Y), strict_implies(Y, X)))) | (strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)) = and(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)), strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(374,plain,
% 4.30/3.04 (strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)) = and(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)), strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[373, 44])).
% 4.30/3.04 tff(375,plain,
% 4.30/3.04 (and(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)), strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))) = strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.04 inference(symmetry,[status(thm)],[374])).
% 4.30/3.04 tff(376,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))) = necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(377,plain,
% 4.30/3.04 (strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))) = necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[376, 20])).
% 4.30/3.04 tff(378,plain,
% 4.30/3.04 (necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))) = strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))),
% 4.30/3.04 inference(symmetry,[status(thm)],[377])).
% 4.30/3.04 tff(379,plain,
% 4.30/3.04 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)) = necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))))),
% 4.30/3.04 inference(quant_inst,[status(thm)],[])).
% 4.30/3.04 tff(380,plain,
% 4.30/3.04 (strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)) = necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.04 inference(unit_resolution,[status(thm)],[379, 20])).
% 4.30/3.04 tff(381,plain,
% 4.30/3.04 (necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))) = strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.04 inference(symmetry,[status(thm)],[380])).
% 4.30/3.04 tff(382,plain,
% 4.30/3.04 (and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))) = and(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)), strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.04 inference(monotonicity,[status(thm)],[381, 378])).
% 4.30/3.04 tff(383,plain,
% 4.30/3.04 (and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))) = strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.05 inference(transitivity,[status(thm)],[382, 375])).
% 4.30/3.05 tff(384,plain,
% 4.30/3.05 (is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) <=> is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[383])).
% 4.30/3.05 tff(385,plain,
% 4.30/3.05 (is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))) <=> is_a_theorem(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[381])).
% 4.30/3.05 tff(386,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))) <=> is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))))),
% 4.30/3.05 inference(symmetry,[status(thm)],[385])).
% 4.30/3.05 tff(387,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(388,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[387, 76])).
% 4.30/3.05 tff(389,plain,
% 4.30/3.05 (is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[388, 386])).
% 4.30/3.05 tff(390,plain,
% 4.30/3.05 (is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))) <=> is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[378])).
% 4.30/3.05 tff(391,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))) <=> is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(symmetry,[status(thm)],[390])).
% 4.30/3.05 tff(392,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), and(Y, X)))) | is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(393,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[392, 76])).
% 4.30/3.05 tff(394,plain,
% 4.30/3.05 (is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[393, 391])).
% 4.30/3.05 tff(395,plain,
% 4.30/3.05 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))))))),
% 4.30/3.05 inference(rewrite,[status(thm)],[])).
% 4.30/3.05 tff(396,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))))))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(397,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[396, 395])).
% 4.30/3.05 tff(398,plain,
% 4.30/3.05 (is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3)))))))) | (~is_a_theorem(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[397, 104])).
% 4.30/3.05 tff(399,plain,
% 4.30/3.05 (is_a_theorem(and(necessarily(implies(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4))), necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), and(X!4, not(and(X!4, not(Y!3))))))))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[398, 394, 389])).
% 4.30/3.05 tff(400,plain,
% 4.30/3.05 (is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[399, 384])).
% 4.30/3.05 tff(401,plain,
% 4.30/3.05 (((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))) | (and(X!4, not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), X!4)))) <=> ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))) | (and(X!4, not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(rewrite,[status(thm)],[])).
% 4.30/3.05 tff(402,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))) | (and(X!4, not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(403,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))) | (and(X!4, not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[402, 401])).
% 4.30/3.05 tff(404,plain,
% 4.30/3.05 ((~is_a_theorem(strict_equiv(and(X!4, not(and(X!4, not(Y!3)))), and(not(and(X!4, not(Y!3))), X!4)))) | (and(X!4, not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[403, 127])).
% 4.30/3.05 tff(405,plain,
% 4.30/3.05 (and(X!4, not(and(X!4, not(Y!3)))) = and(not(and(X!4, not(Y!3))), X!4)),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[404, 400])).
% 4.30/3.05 tff(406,plain,
% 4.30/3.05 (and(not(and(X!4, not(Y!3))), X!4) = and(X!4, not(and(X!4, not(Y!3))))),
% 4.30/3.05 inference(symmetry,[status(thm)],[405])).
% 4.30/3.05 tff(407,plain,
% 4.30/3.05 (and(not(and(not(Y!3), X!4)), X!4) = and(not(and(X!4, not(Y!3))), X!4)),
% 4.30/3.05 inference(monotonicity,[status(thm)],[299])).
% 4.30/3.05 tff(408,plain,
% 4.30/3.05 (and(not(and(not(Y!3), X!4)), X!4) = and(X!4, not(and(X!4, not(Y!3))))),
% 4.30/3.05 inference(transitivity,[status(thm)],[407, 406])).
% 4.30/3.05 tff(409,plain,
% 4.30/3.05 (and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)) = and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[408])).
% 4.30/3.05 tff(410,plain,
% 4.30/3.05 (and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))),
% 4.30/3.05 inference(transitivity,[status(thm)],[409, 372])).
% 4.30/3.05 tff(411,plain,
% 4.30/3.05 (strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3)))) = strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[410, 338])).
% 4.30/3.05 tff(412,plain,
% 4.30/3.05 (strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))) = strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3))))),
% 4.30/3.05 inference(symmetry,[status(thm)],[411])).
% 4.30/3.05 tff(413,plain,
% 4.30/3.05 (strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))) = strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))),
% 4.30/3.05 inference(transitivity,[status(thm)],[412, 303])).
% 4.30/3.05 tff(414,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))) <=> is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[413])).
% 4.30/3.05 tff(415,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4)))) <=> is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.05 inference(symmetry,[status(thm)],[414])).
% 4.30/3.05 tff(416,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3))))) <=> is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[303])).
% 4.30/3.05 tff(417,plain,
% 4.30/3.05 ((~![X: $i, Y: $i, Z: $i] : is_a_theorem(strict_implies(and(and(X, Y), Z), and(X, and(Y, Z))))) | is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3)))))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(418,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(not(Y!3), X!4)), and(X!4, not(Y!3)))))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[417, 295])).
% 4.30/3.05 tff(419,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(and(not(and(not(Y!3), X!4)), X!4), not(Y!3)), and(not(and(X!4, not(Y!3))), and(not(Y!3), X!4))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[418, 416])).
% 4.30/3.05 tff(420,plain,
% 4.30/3.05 (is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[419, 415])).
% 4.30/3.05 tff(421,plain,
% 4.30/3.05 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | ((~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))))),
% 4.30/3.05 inference(rewrite,[status(thm)],[])).
% 4.30/3.05 tff(422,plain,
% 4.30/3.05 ((is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))))) <=> ((~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))))),
% 4.30/3.05 inference(rewrite,[status(thm)],[])).
% 4.30/3.05 tff(423,plain,
% 4.30/3.05 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | ((~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))))),
% 4.30/3.05 inference(monotonicity,[status(thm)],[422])).
% 4.30/3.05 tff(424,plain,
% 4.30/3.05 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))))),
% 4.30/3.05 inference(transitivity,[status(thm)],[423, 421])).
% 4.30/3.05 tff(425,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(426,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[425, 424])).
% 4.30/3.05 tff(427,plain,
% 4.30/3.05 ((~is_a_theorem(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))))) | (~is_a_theorem(strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))))) | is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[426, 104])).
% 4.30/3.05 tff(428,plain,
% 4.30/3.05 (is_a_theorem(and(necessarily(implies(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))), strict_implies(and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))), and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(unit_resolution,[status(thm)],[427, 420, 298])).
% 4.30/3.05 tff(429,plain,
% 4.30/3.05 (is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[428, 277])).
% 4.30/3.05 tff(430,plain,
% 4.30/3.05 (((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) <=> ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(rewrite,[status(thm)],[])).
% 4.30/3.05 tff(431,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | ((~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))),
% 4.30/3.05 inference(quant_inst,[status(thm)],[])).
% 4.30/3.05 tff(432,plain,
% 4.30/3.05 ((~![X: $i, Y: $i] : ((~is_a_theorem(strict_equiv(X, Y))) | (X = Y))) | (~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.05 inference(modus_ponens,[status(thm)],[431, 430])).
% 4.30/3.05 tff(433,plain,
% 4.30/3.05 ((~is_a_theorem(strict_equiv(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))), and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))))) | (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))))),
% 4.30/3.06 inference(unit_resolution,[status(thm)],[432, 127])).
% 4.30/3.06 tff(434,plain,
% 4.30/3.06 (and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))) = and(not(Y!3), and(X!4, not(and(X!4, not(Y!3)))))),
% 4.30/3.06 inference(unit_resolution,[status(thm)],[433, 429])).
% 4.30/3.06 tff(435,plain,
% 4.30/3.06 (and(not(Y!3), and(X!4, not(and(X!4, not(Y!3))))) = and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))),
% 4.30/3.06 inference(symmetry,[status(thm)],[434])).
% 4.30/3.06 tff(436,plain,
% 4.30/3.06 (and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)) = and(and(not(Y!3), X!4), not(and(X!4, not(Y!3))))),
% 4.30/3.06 inference(transitivity,[status(thm)],[372, 435])).
% 4.30/3.06 tff(437,plain,
% 4.30/3.06 (not(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))) = not(and(and(not(Y!3), X!4), not(and(X!4, not(Y!3)))))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[436])).
% 4.30/3.06 tff(438,plain,
% 4.30/3.06 ((~![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) | (implies(and(X!4, not(and(X!4, not(Y!3)))), Y!3) = not(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3))))),
% 4.30/3.06 inference(quant_inst,[status(thm)],[])).
% 4.30/3.06 tff(439,plain,
% 4.30/3.06 (implies(and(X!4, not(and(X!4, not(Y!3)))), Y!3) = not(and(and(X!4, not(and(X!4, not(Y!3)))), not(Y!3)))),
% 4.30/3.06 inference(unit_resolution,[status(thm)],[438, 265])).
% 4.30/3.06 tff(440,plain,
% 4.30/3.06 (implies(and(not(and(X!4, not(Y!3))), X!4), Y!3) = implies(and(X!4, not(and(X!4, not(Y!3)))), Y!3)),
% 4.30/3.06 inference(monotonicity,[status(thm)],[406])).
% 4.30/3.06 tff(441,plain,
% 4.30/3.06 (implies(and(not(and(X!4, not(Y!3))), X!4), Y!3) = implies(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.06 inference(transitivity,[status(thm)],[440, 439, 437, 268, 246])).
% 4.30/3.06 tff(442,plain,
% 4.30/3.06 (necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), Y!3)) = necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[441])).
% 4.30/3.06 tff(443,plain,
% 4.30/3.06 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3) = necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), Y!3)))),
% 4.30/3.06 inference(quant_inst,[status(thm)],[])).
% 4.30/3.06 tff(444,plain,
% 4.30/3.06 (strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3) = necessarily(implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))),
% 4.30/3.06 inference(unit_resolution,[status(thm)],[443, 20])).
% 4.30/3.06 tff(445,plain,
% 4.30/3.06 (strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3) = and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))),
% 4.30/3.06 inference(transitivity,[status(thm)],[444, 442, 23, 172, 142, 245, 49, 133, 46, 25])).
% 4.30/3.06 tff(446,plain,
% 4.30/3.06 (is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3)) <=> is_a_theorem(and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[445])).
% 4.30/3.06 tff(447,plain,
% 4.30/3.06 (is_a_theorem(and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))) <=> is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))),
% 4.30/3.06 inference(symmetry,[status(thm)],[446])).
% 4.30/3.06 tff(448,plain,
% 4.30/3.06 (and(strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4)), strict_implies(and(not(Y!3), X!4), and(not(Y!3), X!4))) = strict_equiv(and(not(Y!3), X!4), and(not(Y!3), X!4))),
% 4.30/3.06 inference(symmetry,[status(thm)],[46])).
% 4.30/3.06 tff(449,plain,
% 4.30/3.06 (and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))) = strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4))),
% 4.30/3.06 inference(transitivity,[status(thm)],[24, 448, 160])).
% 4.30/3.06 tff(450,plain,
% 4.30/3.06 (is_a_theorem(and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))))) <=> is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4)))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[449])).
% 4.30/3.06 tff(451,plain,
% 4.30/3.06 (is_a_theorem(strict_equiv(and(X!4, not(Y!3)), and(not(Y!3), X!4))) <=> is_a_theorem(and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.06 inference(symmetry,[status(thm)],[450])).
% 4.30/3.06 tff(452,plain,
% 4.30/3.06 (is_a_theorem(and(necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4))), necessarily(implies(and(not(Y!3), X!4), and(not(Y!3), X!4)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[110, 451])).
% 4.30/3.06 tff(453,plain,
% 4.30/3.06 (is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[452, 447])).
% 4.30/3.06 tff(454,plain,
% 4.30/3.06 ((~![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) | (implies(X!4, Y!3) = not(and(X!4, not(Y!3))))),
% 4.30/3.06 inference(quant_inst,[status(thm)],[])).
% 4.30/3.06 tff(455,plain,
% 4.30/3.06 (implies(X!4, Y!3) = not(and(X!4, not(Y!3)))),
% 4.30/3.06 inference(unit_resolution,[status(thm)],[454, 265])).
% 4.30/3.06 tff(456,plain,
% 4.30/3.06 (not(and(X!4, not(Y!3))) = implies(X!4, Y!3)),
% 4.30/3.06 inference(symmetry,[status(thm)],[455])).
% 4.30/3.06 tff(457,plain,
% 4.30/3.06 (is_a_theorem(not(and(X!4, not(Y!3)))) <=> is_a_theorem(implies(X!4, Y!3))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[456])).
% 4.30/3.06 tff(458,plain,
% 4.30/3.06 (is_a_theorem(implies(X!4, Y!3)) <=> is_a_theorem(not(and(X!4, not(Y!3))))),
% 4.30/3.06 inference(symmetry,[status(thm)],[457])).
% 4.30/3.06 tff(459,plain,
% 4.30/3.06 ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 4.30/3.06 inference(rewrite,[status(thm)],[])).
% 4.30/3.06 tff(460,plain,
% 4.30/3.06 (($false <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 4.30/3.06 inference(rewrite,[status(thm)],[])).
% 4.30/3.06 tff(461,axiom,(~modus_ponens), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','hilbert_modus_ponens')).
% 4.30/3.06 tff(462,plain,
% 4.30/3.06 (modus_ponens <=> $false),
% 4.30/3.06 inference(iff_false,[status(thm)],[461])).
% 4.30/3.06 tff(463,plain,
% 4.30/3.06 ((modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> ($false <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[462])).
% 4.30/3.06 tff(464,plain,
% 4.30/3.06 ((modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 4.30/3.06 inference(transitivity,[status(thm)],[463, 460])).
% 4.30/3.06 tff(465,plain,
% 4.30/3.06 ((modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 4.30/3.06 inference(rewrite,[status(thm)],[])).
% 4.30/3.06 tff(466,plain,
% 4.30/3.06 ((modus_ponens <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))) <=> (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 4.30/3.06 inference(rewrite,[status(thm)],[])).
% 4.30/3.06 tff(467,axiom,(modus_ponens <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+0.ax','modus_ponens')).
% 4.30/3.06 tff(468,plain,
% 4.30/3.06 (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[467, 466])).
% 4.30/3.06 tff(469,plain,
% 4.30/3.06 (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[468, 465])).
% 4.30/3.06 tff(470,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[469, 464])).
% 4.30/3.06 tff(471,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[470, 459])).
% 4.30/3.06 tff(472,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[471, 459])).
% 4.30/3.06 tff(473,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[472, 459])).
% 4.30/3.06 tff(474,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[473, 459])).
% 4.30/3.06 tff(475,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[474, 459])).
% 4.30/3.06 tff(476,plain,
% 4.30/3.06 (~![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[475, 459])).
% 4.30/3.06 tff(477,plain,(
% 4.30/3.06 ~(is_a_theorem(Y!3) | (~(is_a_theorem(X!4) & is_a_theorem(implies(X!4, Y!3)))))),
% 4.30/3.06 inference(skolemize,[status(sab)],[476])).
% 4.30/3.06 tff(478,plain,
% 4.30/3.06 (is_a_theorem(X!4) & is_a_theorem(implies(X!4, Y!3))),
% 4.30/3.06 inference(or_elim,[status(thm)],[477])).
% 4.30/3.06 tff(479,plain,
% 4.30/3.06 (is_a_theorem(implies(X!4, Y!3))),
% 4.30/3.06 inference(and_elim,[status(thm)],[478])).
% 4.30/3.06 tff(480,plain,
% 4.30/3.06 (is_a_theorem(not(and(X!4, not(Y!3))))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[479, 458])).
% 4.30/3.06 tff(481,plain,
% 4.30/3.06 (is_a_theorem(X!4)),
% 4.30/3.06 inference(and_elim,[status(thm)],[478])).
% 4.30/3.06 tff(482,plain,
% 4.30/3.06 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | ((~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))) | is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))) | is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.06 inference(rewrite,[status(thm)],[])).
% 4.30/3.06 tff(483,plain,
% 4.30/3.06 ((is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3)))))) <=> ((~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))) | is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.06 inference(rewrite,[status(thm)],[])).
% 4.30/3.06 tff(484,plain,
% 4.30/3.06 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | ((~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))) | is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))))),
% 4.30/3.06 inference(monotonicity,[status(thm)],[483])).
% 4.30/3.06 tff(485,plain,
% 4.30/3.06 (((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))) | is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)))),
% 4.30/3.06 inference(transitivity,[status(thm)],[484, 482])).
% 4.30/3.06 tff(486,plain,
% 4.30/3.06 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))))),
% 4.30/3.06 inference(quant_inst,[status(thm)],[])).
% 4.30/3.06 tff(487,plain,
% 4.30/3.06 ((~![X: $i, Y: $i] : (is_a_theorem(and(X, Y)) | (~is_a_theorem(Y)) | (~is_a_theorem(X)))) | (~is_a_theorem(X!4)) | (~is_a_theorem(not(and(X!4, not(Y!3))))) | is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.06 inference(modus_ponens,[status(thm)],[486, 485])).
% 4.30/3.06 tff(488,plain,
% 4.30/3.06 (is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))),
% 4.30/3.06 inference(unit_resolution,[status(thm)],[487, 481, 104, 480])).
% 4.30/3.06 tff(489,plain,
% 4.30/3.06 (^[X: $i, Y: $i] : refl((is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X))) <=> (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X))))),
% 4.30/3.06 inference(bind,[status(th)],[])).
% 4.30/3.06 tff(490,plain,
% 4.30/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))),
% 4.37/3.06 inference(quant_intro,[status(thm)],[489])).
% 4.37/3.06 tff(491,plain,
% 4.37/3.06 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))) <=> (~((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y)))))), ((~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))) <=> (~(~((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y)))))))), rewrite((~(~((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y)))))) <=> ((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y))))), ((~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))) <=> ((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y)))))), ((is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))) <=> (is_a_theorem(Y) | ((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y))))))), rewrite((is_a_theorem(Y) | ((~is_a_theorem(X)) | (~is_a_theorem(strict_implies(X, Y))))) <=> (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))), ((is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))) <=> (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))))),
% 4.37/3.06 inference(bind,[status(th)],[])).
% 4.37/3.06 tff(492,plain,
% 4.37/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))),
% 4.37/3.06 inference(quant_intro,[status(thm)],[491])).
% 4.37/3.06 tff(493,plain,
% 4.37/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(rewrite,[status(thm)],[])).
% 4.37/3.06 tff(494,plain,
% 4.37/3.06 (($true <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(rewrite,[status(thm)],[])).
% 4.37/3.06 tff(495,axiom,(modus_ponens_strict_implies), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+4.ax','s1_0_modus_ponens_strict_implies')).
% 4.37/3.06 tff(496,plain,
% 4.37/3.06 (modus_ponens_strict_implies <=> $true),
% 4.37/3.06 inference(iff_true,[status(thm)],[495])).
% 4.37/3.06 tff(497,plain,
% 4.37/3.06 ((modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))) <=> ($true <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))))),
% 4.37/3.06 inference(monotonicity,[status(thm)],[496])).
% 4.37/3.06 tff(498,plain,
% 4.37/3.06 ((modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(transitivity,[status(thm)],[497, 494])).
% 4.37/3.06 tff(499,plain,
% 4.37/3.06 ((modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))) <=> (modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))))),
% 4.37/3.06 inference(rewrite,[status(thm)],[])).
% 4.37/3.06 tff(500,plain,
% 4.37/3.06 ((modus_ponens_strict_implies <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))) => is_a_theorem(Y))) <=> (modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))))))),
% 4.37/3.06 inference(rewrite,[status(thm)],[])).
% 4.37/3.06 tff(501,axiom,(modus_ponens_strict_implies <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(strict_implies(X, Y))) => is_a_theorem(Y))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','modus_ponens_strict_implies')).
% 4.37/3.06 tff(502,plain,
% 4.37/3.06 (modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[501, 500])).
% 4.37/3.06 tff(503,plain,
% 4.37/3.06 (modus_ponens_strict_implies <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[502, 499])).
% 4.37/3.06 tff(504,plain,
% 4.37/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[503, 498])).
% 4.37/3.06 tff(505,plain,
% 4.37/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[504, 493])).
% 4.37/3.06 tff(506,plain,(
% 4.37/3.06 ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(strict_implies(X, Y)))))),
% 4.37/3.06 inference(skolemize,[status(sab)],[505])).
% 4.37/3.06 tff(507,plain,
% 4.37/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[506, 492])).
% 4.37/3.06 tff(508,plain,
% 4.37/3.06 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[507, 490])).
% 4.37/3.06 tff(509,plain,
% 4.37/3.06 (~is_a_theorem(Y!3)),
% 4.37/3.06 inference(or_elim,[status(thm)],[477])).
% 4.37/3.06 tff(510,plain,
% 4.37/3.06 (((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | (is_a_theorem(Y!3) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | is_a_theorem(Y!3) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))))),
% 4.37/3.06 inference(rewrite,[status(thm)],[])).
% 4.37/3.06 tff(511,plain,
% 4.37/3.06 ((is_a_theorem(Y!3) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4)))) <=> (is_a_theorem(Y!3) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))))),
% 4.37/3.06 inference(rewrite,[status(thm)],[])).
% 4.37/3.06 tff(512,plain,
% 4.37/3.06 (((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | (is_a_theorem(Y!3) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | (is_a_theorem(Y!3) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3)))))),
% 4.37/3.06 inference(monotonicity,[status(thm)],[511])).
% 4.37/3.06 tff(513,plain,
% 4.37/3.06 (((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | (is_a_theorem(Y!3) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | is_a_theorem(Y!3) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))))),
% 4.37/3.06 inference(transitivity,[status(thm)],[512, 510])).
% 4.37/3.06 tff(514,plain,
% 4.37/3.06 ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | (is_a_theorem(Y!3) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))))),
% 4.37/3.06 inference(quant_inst,[status(thm)],[])).
% 4.37/3.06 tff(515,plain,
% 4.37/3.06 ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(strict_implies(X, Y))) | (~is_a_theorem(X)))) | is_a_theorem(Y!3) | (~is_a_theorem(and(not(and(X!4, not(Y!3))), X!4))) | (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3)))),
% 4.37/3.06 inference(modus_ponens,[status(thm)],[514, 513])).
% 4.37/3.06 tff(516,plain,
% 4.37/3.06 (~is_a_theorem(strict_implies(and(not(and(X!4, not(Y!3))), X!4), Y!3))),
% 4.37/3.06 inference(unit_resolution,[status(thm)],[515, 509, 508, 488])).
% 4.37/3.08 tff(517,plain,
% 4.37/3.08 ($false),
% 4.37/3.08 inference(unit_resolution,[status(thm)],[516, 453])).
% 4.37/3.08 % SZS output end Proof
%------------------------------------------------------------------------------