ITP001 Axioms: ITP003_5.ax


%------------------------------------------------------------------------------
% File     : ITP003_5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : bool_2.ax [Gau20]
%          : HL4003_5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  394 (  77 unt; 102 typ;   0 def)
%            Number of atoms       : 2665 ( 179 equ)
%            Maximal formula atoms :   19 (   6 avg)
%            Number of connectives :  908 (  46   ~;  61   |; 182   &)
%                                         ( 151 <=>; 468  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   6 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of FOOLs       : 1511 (1511 fml;   0 var)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :  197 (  89   >; 108   *;   0   +;   0  <<)
%            Number of predicates  :   17 (  14 usr;   7 prp; 0-2 aty)
%            Number of functors    :  101 ( 101 usr;  12 con; 0-5 aty)
%            Number of variables   :  892 ( 836   !;  56   ?; 892   :)
% SPC      : TF0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_ty_2Ebool_2Eitself,type,
    ty_2Ebool_2Eitself: del > del ).

tff(tp_c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: del > $i ).

tff(mem_c_2Ebool_2E_21,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ).

tff(ax_all_p,axiom,
    ! [A: del,Q: $i] :
      ( mem(Q,arr(A,bool))
     => ( p(ap(c_2Ebool_2E_21(A),Q))
      <=> ! [X: $i] :
            ( mem(X,A)
           => p(ap(Q,X)) ) ) ) ).

tff(tp_c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $i ).

tff(mem_c_2Ebool_2E_2F_5C,axiom,
    mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).

tff(stp_fo_c_2Ebool_2E_2F_5C,type,
    fo__c_2Ebool_2E_2F_5C: ( tp__o * tp__o ) > tp__o ).

tff(stp_eq_fo_c_2Ebool_2E_2F_5C,axiom,
    ! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Ebool_2E_2F_5C(X0,X1)) = ap(ap(c_2Ebool_2E_2F_5C,inj__o(X0)),inj__o(X1)) ) ).

tff(ax_and_p,axiom,
    ! [Q: $i] :
      ( mem(Q,bool)
     => ! [R: $i] :
          ( mem(R,bool)
         => ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
          <=> ( p(Q)
              & p(R) ) ) ) ) ).

tff(tp_c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: del > $i ).

tff(mem_c_2Ebool_2E_3F,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ).

tff(ax_ex_p,axiom,
    ! [A: del,Q: $i] :
      ( mem(Q,arr(A,bool))
     => ( p(ap(c_2Ebool_2E_3F(A),Q))
      <=> ? [X: $i] :
            ( mem(X,A)
            & p(ap(Q,X)) ) ) ) ).

tff(tp_c_2Ebool_2E_3F_21,type,
    c_2Ebool_2E_3F_21: del > $i ).

tff(mem_c_2Ebool_2E_3F_21,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2E_3F_21(A_27a),arr(arr(A_27a,bool),bool)) ).

tff(tp_c_2Ebool_2EARB,type,
    c_2Ebool_2EARB: del > $i ).

tff(mem_c_2Ebool_2EARB,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2EARB(A_27a),A_27a) ).

tff(tp_c_2Ebool_2EBOUNDED,type,
    c_2Ebool_2EBOUNDED: $i ).

tff(mem_c_2Ebool_2EBOUNDED,axiom,
    mem(c_2Ebool_2EBOUNDED,arr(bool,bool)) ).

tff(stp_fo_c_2Ebool_2EBOUNDED,type,
    fo__c_2Ebool_2EBOUNDED: tp__o > tp__o ).

tff(stp_eq_fo_c_2Ebool_2EBOUNDED,axiom,
    ! [X0: tp__o] : ( inj__o(fo__c_2Ebool_2EBOUNDED(X0)) = ap(c_2Ebool_2EBOUNDED,inj__o(X0)) ) ).

tff(tp_c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: del > $i ).

tff(mem_c_2Ebool_2ECOND,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2ECOND(A_27a),arr(bool,arr(A_27a,arr(A_27a,A_27a)))) ).

tff(tp_c_2Ebool_2EDATATYPE,type,
    c_2Ebool_2EDATATYPE: del > $i ).

tff(mem_c_2Ebool_2EDATATYPE,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2EDATATYPE(A_27a),arr(A_27a,bool)) ).

tff(tp_c_2Ebool_2EF,type,
    c_2Ebool_2EF: $i ).

tff(mem_c_2Ebool_2EF,axiom,
    mem(c_2Ebool_2EF,bool) ).

tff(stp_fo_c_2Ebool_2EF,type,
    fo__c_2Ebool_2EF: tp__o ).

tff(stp_eq_fo_c_2Ebool_2EF,axiom,
    inj__o(fo__c_2Ebool_2EF) = c_2Ebool_2EF ).

tff(ax_false_p,axiom,
    ~ p(c_2Ebool_2EF) ).

tff(tp_c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: del > $i ).

tff(mem_c_2Ebool_2EIN,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2EIN(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool))) ).

tff(tp_c_2Ebool_2ELET,type,
    c_2Ebool_2ELET: ( del * del ) > $i ).

tff(mem_c_2Ebool_2ELET,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2ELET(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(A_27a,A_27b))) ).

tff(tp_c_2Ebool_2EONE__ONE,type,
    c_2Ebool_2EONE__ONE: ( del * del ) > $i ).

tff(mem_c_2Ebool_2EONE__ONE,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2EONE__ONE(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)) ).

tff(tp_c_2Ebool_2EONTO,type,
    c_2Ebool_2EONTO: ( del * del ) > $i ).

tff(mem_c_2Ebool_2EONTO,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2EONTO(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)) ).

tff(tp_c_2Ebool_2ERES__ABSTRACT,type,
    c_2Ebool_2ERES__ABSTRACT: ( del * del ) > $i ).

tff(mem_c_2Ebool_2ERES__ABSTRACT,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))) ).

tff(tp_c_2Ebool_2ERES__EXISTS,type,
    c_2Ebool_2ERES__EXISTS: del > $i ).

tff(mem_c_2Ebool_2ERES__EXISTS,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2ERES__EXISTS(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).

tff(tp_c_2Ebool_2ERES__EXISTS__UNIQUE,type,
    c_2Ebool_2ERES__EXISTS__UNIQUE: del > $i ).

tff(mem_c_2Ebool_2ERES__EXISTS__UNIQUE,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).

tff(tp_c_2Ebool_2ERES__FORALL,type,
    c_2Ebool_2ERES__FORALL: del > $i ).

tff(mem_c_2Ebool_2ERES__FORALL,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2ERES__FORALL(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).

tff(tp_c_2Ebool_2ERES__SELECT,type,
    c_2Ebool_2ERES__SELECT: del > $i ).

tff(mem_c_2Ebool_2ERES__SELECT,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2ERES__SELECT(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),A_27a))) ).

tff(tp_c_2Ebool_2ET,type,
    c_2Ebool_2ET: $i ).

tff(mem_c_2Ebool_2ET,axiom,
    mem(c_2Ebool_2ET,bool) ).

tff(stp_fo_c_2Ebool_2ET,type,
    fo__c_2Ebool_2ET: tp__o ).

tff(stp_eq_fo_c_2Ebool_2ET,axiom,
    inj__o(fo__c_2Ebool_2ET) = c_2Ebool_2ET ).

tff(ax_true_p,axiom,
    p(c_2Ebool_2ET) ).

tff(tp_c_2Ebool_2ETYPE__DEFINITION,type,
    c_2Ebool_2ETYPE__DEFINITION: ( del * del ) > $i ).

tff(mem_c_2Ebool_2ETYPE__DEFINITION,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,A_27a),bool))) ).

tff(tp_c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $i ).

tff(mem_c_2Ebool_2E_5C_2F,axiom,
    mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).

tff(stp_fo_c_2Ebool_2E_5C_2F,type,
    fo__c_2Ebool_2E_5C_2F: ( tp__o * tp__o ) > tp__o ).

tff(stp_eq_fo_c_2Ebool_2E_5C_2F,axiom,
    ! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Ebool_2E_5C_2F(X0,X1)) = ap(ap(c_2Ebool_2E_5C_2F,inj__o(X0)),inj__o(X1)) ) ).

tff(ax_or_p,axiom,
    ! [Q: $i] :
      ( mem(Q,bool)
     => ! [R: $i] :
          ( mem(R,bool)
         => ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
          <=> ( p(Q)
              | p(R) ) ) ) ) ).

tff(tp_c_2Ebool_2Eitself__case,type,
    c_2Ebool_2Eitself__case: ( del * del ) > $i ).

tff(mem_c_2Ebool_2Eitself__case,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2Eitself__case(A_27a,A_27b),arr(ty_2Ebool_2Eitself(A_27a),arr(A_27b,A_27b))) ).

tff(tp_c_2Ebool_2Eliteral__case,type,
    c_2Ebool_2Eliteral__case: ( del * del ) > $i ).

tff(mem_c_2Ebool_2Eliteral__case,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Ebool_2Eliteral__case(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(A_27a,A_27b))) ).

tff(tp_c_2Ebool_2Ethe__value,type,
    c_2Ebool_2Ethe__value: del > $i ).

tff(mem_c_2Ebool_2Ethe__value,axiom,
    ! [A_27a: del] : mem(c_2Ebool_2Ethe__value(A_27a),ty_2Ebool_2Eitself(A_27a)) ).

tff(tp_c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $i ).

tff(mem_c_2Ebool_2E_7E,axiom,
    mem(c_2Ebool_2E_7E,arr(bool,bool)) ).

tff(stp_fo_c_2Ebool_2E_7E,type,
    fo__c_2Ebool_2E_7E: tp__o > tp__o ).

tff(stp_eq_fo_c_2Ebool_2E_7E,axiom,
    ! [X0: tp__o] : ( inj__o(fo__c_2Ebool_2E_7E(X0)) = ap(c_2Ebool_2E_7E,inj__o(X0)) ) ).

tff(ax_neg_p,axiom,
    ! [Q: $i] :
      ( mem(Q,bool)
     => ( p(ap(c_2Ebool_2E_7E,Q))
      <=> ~ p(Q) ) ) ).

tff(ax_thm_2Ebool_2ET__DEF,axiom,
    ( $true
  <=> ( i(bool) = i(bool) ) ) ).

tff(lamtp_f1,type,
    f1: del > $i ).

tff(lameq_f1,axiom,
    ! [A_27a: del,V0P: $i] : ( ap(f1(A_27a),V0P) = ap(ap(c_2Emin_2E_3D(arr(A_27a,bool)),V0P),k(A_27a,c_2Ebool_2ET)) ) ).

tff(ax_thm_2Ebool_2EFORALL__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2E_21(A_27a) = f1(A_27a) ) ).

tff(lamtp_f2,type,
    f2: del > $i ).

tff(lameq_f2,axiom,
    ! [A_27a: del,V0P: $i] : ( ap(f2(A_27a),V0P) = ap(V0P,ap(c_2Emin_2E_40(A_27a),V0P)) ) ).

tff(ax_thm_2Ebool_2EEXISTS__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2E_3F(A_27a) = f2(A_27a) ) ).

tff(lamtp_f3,type,
    f3: ( tp__o * tp__o ) > $i ).

tff(lameq_f3,axiom,
    ! [V1t2: tp__o,V0t1: tp__o,V2t: tp__o] : ( ap(f3(V1t2,V0t1),inj__o(V2t)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V0t1)),ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V1t2)),inj__o(V2t)))),inj__o(V2t)) ) ).

tff(lamtp_f4,type,
    f4: tp__o > $i ).

tff(lameq_f4,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] : ( ap(f4(V0t1),inj__o(V1t2)) = ap(c_2Ebool_2E_21(bool),f3(V1t2,V0t1)) ) ).

tff(lamtp_f5,type,
    f5: $i ).

tff(lameq_f5,axiom,
    ! [V0t1: tp__o] : ( ap(f5,inj__o(V0t1)) = f4(V0t1) ) ).

tff(ax_thm_2Ebool_2EAND__DEF,axiom,
    c_2Ebool_2E_2F_5C = f5 ).

tff(lamtp_f6,type,
    f6: ( tp__o * tp__o ) > $i ).

tff(lameq_f6,axiom,
    ! [V0t1: tp__o,V1t2: tp__o,V2t: tp__o] : ( ap(f6(V0t1,V1t2),inj__o(V2t)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V0t1)),inj__o(V2t))),ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V1t2)),inj__o(V2t))),inj__o(V2t))) ) ).

tff(lamtp_f7,type,
    f7: tp__o > $i ).

tff(lameq_f7,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] : ( ap(f7(V0t1),inj__o(V1t2)) = ap(c_2Ebool_2E_21(bool),f6(V0t1,V1t2)) ) ).

tff(lamtp_f8,type,
    f8: $i ).

tff(lameq_f8,axiom,
    ! [V0t1: tp__o] : ( ap(f8,inj__o(V0t1)) = f7(V0t1) ) ).

tff(ax_thm_2Ebool_2EOR__DEF,axiom,
    c_2Ebool_2E_5C_2F = f8 ).

tff(ax_thm_2Ebool_2EF__DEF,axiom,
    ( $false
  <=> ! [V0t: tp__o] : p(inj__o(V0t)) ) ).

tff(lamtp_f9,type,
    f9: $i ).

tff(lameq_f9,axiom,
    ! [V0t: tp__o] : ( ap(f9,inj__o(V0t)) = ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V0t)),inj__o(fo__c_2Ebool_2EF)) ) ).

tff(ax_thm_2Ebool_2ENOT__DEF,axiom,
    c_2Ebool_2E_7E = f9 ).

tff(lamtp_f10,type,
    f10: ( del * $i * $i ) > $i ).

tff(lameq_f10,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1x: $i] :
          ( mem(V1x,A_27a)
         => ! [V2y: $i] : ( ap(f10(A_27a,V0P,V1x),V2y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V1x)),ap(V0P,V2y))),ap(ap(c_2Emin_2E_3D(A_27a),V1x),V2y)) ) ) ) ).

tff(lamtp_f11,type,
    f11: ( del * $i ) > $i ).

tff(lameq_f11,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1x: $i] : ( ap(f11(A_27a,V0P),V1x) = ap(c_2Ebool_2E_21(A_27a),f10(A_27a,V0P,V1x)) ) ) ).

tff(lamtp_f12,type,
    f12: del > $i ).

tff(lameq_f12,axiom,
    ! [A_27a: del,V0P: $i] : ( ap(f12(A_27a),V0P) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_3F(A_27a),V0P)),ap(c_2Ebool_2E_21(A_27a),f11(A_27a,V0P))) ) ).

tff(ax_thm_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2E_3F_21(A_27a) = f12(A_27a) ) ).

tff(lamtp_f13,type,
    f13: ( del * del * $i ) > $i ).

tff(lameq_f13,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1x: $i] : ( ap(f13(A_27b,A_27a,V0f),V1x) = ap(V0f,V1x) ) ) ).

tff(lamtp_f14,type,
    f14: ( del * del ) > $i ).

tff(lameq_f14,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] : ( ap(f14(A_27b,A_27a),V0f) = f13(A_27b,A_27a,V0f) ) ).

tff(ax_thm_2Ebool_2ELET__DEF,axiom,
    ! [A_27a: del,A_27b: del] : ( c_2Ebool_2ELET(A_27a,A_27b) = f14(A_27b,A_27a) ) ).

tff(lamtp_f15,type,
    f15: ( del * $i * $i * tp__o ) > $i ).

tff(lameq_f15,axiom,
    ! [A_27a: del,V1t1: $i] :
      ( mem(V1t1,A_27a)
     => ! [V2t2: $i] :
          ( mem(V2t2,A_27a)
         => ! [V0t: tp__o,V3x: $i] : ( ap(f15(A_27a,V1t1,V2t2,V0t),V3x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(bool),inj__o(V0t)),inj__o(fo__c_2Ebool_2ET))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V1t1))),ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(bool),inj__o(V0t)),inj__o(fo__c_2Ebool_2EF))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V2t2))) ) ) ) ).

tff(lamtp_f16,type,
    f16: ( del * tp__o * $i ) > $i ).

tff(lameq_f16,axiom,
    ! [A_27a: del,V0t: tp__o,V1t1: $i] :
      ( mem(V1t1,A_27a)
     => ! [V2t2: $i] : ( ap(f16(A_27a,V0t,V1t1),V2t2) = ap(c_2Emin_2E_40(A_27a),f15(A_27a,V1t1,V2t2,V0t)) ) ) ).

tff(lamtp_f17,type,
    f17: ( del * tp__o ) > $i ).

tff(lameq_f17,axiom,
    ! [A_27a: del,V0t: tp__o,V1t1: $i] : ( ap(f17(A_27a,V0t),V1t1) = f16(A_27a,V0t,V1t1) ) ).

tff(lamtp_f18,type,
    f18: del > $i ).

tff(lameq_f18,axiom,
    ! [A_27a: del,V0t: tp__o] : ( ap(f18(A_27a),inj__o(V0t)) = f17(A_27a,V0t) ) ).

tff(ax_thm_2Ebool_2ECOND__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2ECOND(A_27a) = f18(A_27a) ) ).

tff(lamtp_f19,type,
    f19: ( del * del * $i * $i ) > $i ).

tff(lameq_f19,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1x1: $i] :
          ( mem(V1x1,A_27a)
         => ! [V2x2: $i] : ( ap(f19(A_27b,A_27a,V0f,V1x1),V2x2) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(A_27b),ap(V0f,V1x1)),ap(V0f,V2x2))),ap(ap(c_2Emin_2E_3D(A_27a),V1x1),V2x2)) ) ) ) ).

tff(lamtp_f20,type,
    f20: ( del * del * $i ) > $i ).

tff(lameq_f20,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1x1: $i] : ( ap(f20(A_27b,A_27a,V0f),V1x1) = ap(c_2Ebool_2E_21(A_27a),f19(A_27b,A_27a,V0f,V1x1)) ) ) ).

tff(lamtp_f21,type,
    f21: ( del * del ) > $i ).

tff(lameq_f21,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] : ( ap(f21(A_27b,A_27a),V0f) = ap(c_2Ebool_2E_21(A_27a),f20(A_27b,A_27a,V0f)) ) ).

tff(ax_thm_2Ebool_2EONE__ONE__DEF,axiom,
    ! [A_27a: del,A_27b: del] : ( c_2Ebool_2EONE__ONE(A_27a,A_27b) = f21(A_27b,A_27a) ) ).

tff(lamtp_f22,type,
    f22: ( del * del * $i * $i ) > $i ).

tff(lameq_f22,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1y: $i] :
          ( mem(V1y,A_27b)
         => ! [V2x: $i] : ( ap(f22(A_27b,A_27a,V0f,V1y),V2x) = ap(ap(c_2Emin_2E_3D(A_27b),V1y),ap(V0f,V2x)) ) ) ) ).

tff(lamtp_f23,type,
    f23: ( del * del * $i ) > $i ).

tff(lameq_f23,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1y: $i] : ( ap(f23(A_27a,A_27b,V0f),V1y) = ap(c_2Ebool_2E_3F(A_27a),f22(A_27b,A_27a,V0f,V1y)) ) ) ).

tff(lamtp_f24,type,
    f24: ( del * del ) > $i ).

tff(lameq_f24,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] : ( ap(f24(A_27a,A_27b),V0f) = ap(c_2Ebool_2E_21(A_27b),f23(A_27a,A_27b,V0f)) ) ).

tff(ax_thm_2Ebool_2EONTO__DEF,axiom,
    ! [A_27a: del,A_27b: del] : ( c_2Ebool_2EONTO(A_27a,A_27b) = f24(A_27a,A_27b) ) ).

tff(lamtp_f25,type,
    f25: ( del * del * $i * $i ) > $i ).

tff(lameq_f25,axiom,
    ! [A_27a: del,A_27b: del,V1rep: $i] :
      ( mem(V1rep,arr(A_27b,A_27a))
     => ! [V2x_27: $i] :
          ( mem(V2x_27,A_27b)
         => ! [V3x_27_27: $i] : ( ap(f25(A_27a,A_27b,V1rep,V2x_27),V3x_27_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(A_27a),ap(V1rep,V2x_27)),ap(V1rep,V3x_27_27))),ap(ap(c_2Emin_2E_3D(A_27b),V2x_27),V3x_27_27)) ) ) ) ).

tff(lamtp_f26,type,
    f26: ( del * del * $i ) > $i ).

tff(lameq_f26,axiom,
    ! [A_27a: del,A_27b: del,V1rep: $i] :
      ( mem(V1rep,arr(A_27b,A_27a))
     => ! [V2x_27: $i] : ( ap(f26(A_27a,A_27b,V1rep),V2x_27) = ap(c_2Ebool_2E_21(A_27b),f25(A_27a,A_27b,V1rep,V2x_27)) ) ) ).

tff(lamtp_f27,type,
    f27: ( del * del * $i * $i ) > $i ).

tff(lameq_f27,axiom,
    ! [A_27a: del,A_27b: del,V1rep: $i] :
      ( mem(V1rep,arr(A_27b,A_27a))
     => ! [V4x: $i] :
          ( mem(V4x,A_27a)
         => ! [V5x_27: $i] : ( ap(f27(A_27a,A_27b,V1rep,V4x),V5x_27) = ap(ap(c_2Emin_2E_3D(A_27a),V4x),ap(V1rep,V5x_27)) ) ) ) ).

tff(lamtp_f28,type,
    f28: ( del * del * $i * $i ) > $i ).

tff(lameq_f28,axiom,
    ! [A_27b: del,A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1rep: $i] :
          ( mem(V1rep,arr(A_27b,A_27a))
         => ! [V4x: $i] : ( ap(f28(A_27b,A_27a,V0P,V1rep),V4x) = ap(ap(c_2Emin_2E_3D(bool),ap(V0P,V4x)),ap(c_2Ebool_2E_3F(A_27b),f27(A_27a,A_27b,V1rep,V4x))) ) ) ) ).

tff(lamtp_f29,type,
    f29: ( del * del * $i ) > $i ).

tff(lameq_f29,axiom,
    ! [A_27b: del,A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1rep: $i] : ( ap(f29(A_27b,A_27a,V0P),V1rep) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(A_27b),f26(A_27a,A_27b,V1rep))),ap(c_2Ebool_2E_21(A_27a),f28(A_27b,A_27a,V0P,V1rep))) ) ) ).

tff(lamtp_f30,type,
    f30: ( del * del ) > $i ).

tff(lameq_f30,axiom,
    ! [A_27b: del,A_27a: del,V0P: $i] : ( ap(f30(A_27b,A_27a),V0P) = f29(A_27b,A_27a,V0P) ) ).

tff(ax_thm_2Ebool_2ETYPE__DEFINITION,axiom,
    ! [A_27a: del,A_27b: del] : ( c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b) = f30(A_27b,A_27a) ) ).

tff(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
    ! [V0t: tp__o] :
      ( ( p(inj__o(V0t))
      <=> $true )
      | ( p(inj__o(V0t))
      <=> $false ) ) ).

tff(lamtp_f31,type,
    f31: ( del * del * $i ) > $i ).

tff(lameq_f31,axiom,
    ! [A_27b: del,A_27a: del,V0t: $i] :
      ( mem(V0t,arr(A_27a,A_27b))
     => ! [V1x: $i] : ( ap(f31(A_27b,A_27a,V0t),V1x) = ap(V0t,V1x) ) ) ).

tff(ax_thm_2Ebool_2EETA__AX,axiom,
    ! [A_27a: del,A_27b: del,V0t: $i] :
      ( mem(V0t,arr(A_27a,A_27b))
     => ( f31(A_27b,A_27a,V0t) = V0t ) ) ).

tff(ax_thm_2Ebool_2ESELECT__AX,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1x: $i] :
          ( mem(V1x,A_27a)
         => ( p(ap(V0P,V1x))
           => p(ap(V0P,ap(c_2Emin_2E_40(A_27a),V0P))) ) ) ) ).

tff(stp_i,type,
    tp__i: $tType ).

tff(stp_inj_i,type,
    inj__i: tp__i > $i ).

tff(stp_surj_i,type,
    surj__i: $i > tp__i ).

tff(stp_inj_surj_i,axiom,
    ! [X: tp__i] : ( surj__i(inj__i(X)) = X ) ).

tff(stp_inj_mem_i,axiom,
    ! [X: tp__i] : mem(inj__i(X),ind) ).

tff(stp_iso_mem_i,axiom,
    ! [X: $i] :
      ( mem(X,ind)
     => ( X = inj__i(surj__i(X)) ) ) ).

tff(ax_thm_2Ebool_2EINFINITY__AX,axiom,
    ? [V0f: $i] :
      ( mem(V0f,arr(ind,ind))
      & p(ap(c_2Ebool_2EONE__ONE(ind,ind),V0f))
      & ~ p(ap(c_2Ebool_2EONTO(ind,ind),V0f)) ) ).

tff(ax_thm_2Ebool_2Eliteral__case__DEF,axiom,
    ! [A_27a: del,A_27b: del] : ( c_2Ebool_2Eliteral__case(A_27a,A_27b) = f14(A_27b,A_27a) ) ).

tff(lamtp_f32,type,
    f32: ( del * $i ) > $i ).

tff(lameq_f32,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ! [V1f: $i] : ( ap(f32(A_27a,V0x),V1f) = ap(V1f,V0x) ) ) ).

tff(lamtp_f33,type,
    f33: del > $i ).

tff(lameq_f33,axiom,
    ! [A_27a: del,V0x: $i] : ( ap(f33(A_27a),V0x) = f32(A_27a,V0x) ) ).

tff(ax_thm_2Ebool_2EIN__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2EIN(A_27a) = f33(A_27a) ) ).

tff(lamtp_f34,type,
    f34: ( del * $i * $i ) > $i ).

tff(lameq_f34,axiom,
    ! [A_27a: del,V0p: $i] :
      ( mem(V0p,arr(A_27a,bool))
     => ! [V1m: $i] :
          ( mem(V1m,arr(A_27a,bool))
         => ! [V2x: $i] : ( ap(f34(A_27a,V0p,V1m),V2x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p)),ap(V1m,V2x)) ) ) ) ).

tff(lamtp_f35,type,
    f35: ( del * $i ) > $i ).

tff(lameq_f35,axiom,
    ! [A_27a: del,V0p: $i] :
      ( mem(V0p,arr(A_27a,bool))
     => ! [V1m: $i] : ( ap(f35(A_27a,V0p),V1m) = ap(c_2Ebool_2E_21(A_27a),f34(A_27a,V0p,V1m)) ) ) ).

tff(lamtp_f36,type,
    f36: del > $i ).

tff(lameq_f36,axiom,
    ! [A_27a: del,V0p: $i] : ( ap(f36(A_27a),V0p) = f35(A_27a,V0p) ) ).

tff(ax_thm_2Ebool_2ERES__FORALL__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2ERES__FORALL(A_27a) = f36(A_27a) ) ).

tff(lamtp_f37,type,
    f37: ( del * $i * $i ) > $i ).

tff(lameq_f37,axiom,
    ! [A_27a: del,V0p: $i] :
      ( mem(V0p,arr(A_27a,bool))
     => ! [V1m: $i] :
          ( mem(V1m,arr(A_27a,bool))
         => ! [V2x: $i] : ( ap(f37(A_27a,V0p,V1m),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p)),ap(V1m,V2x)) ) ) ) ).

tff(lamtp_f38,type,
    f38: ( del * $i ) > $i ).

tff(lameq_f38,axiom,
    ! [A_27a: del,V0p: $i] :
      ( mem(V0p,arr(A_27a,bool))
     => ! [V1m: $i] : ( ap(f38(A_27a,V0p),V1m) = ap(c_2Ebool_2E_3F(A_27a),f37(A_27a,V0p,V1m)) ) ) ).

tff(lamtp_f39,type,
    f39: del > $i ).

tff(lameq_f39,axiom,
    ! [A_27a: del,V0p: $i] : ( ap(f39(A_27a),V0p) = f38(A_27a,V0p) ) ).

tff(ax_thm_2Ebool_2ERES__EXISTS__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2ERES__EXISTS(A_27a) = f39(A_27a) ) ).

tff(lamtp_f40,type,
    f40: ( del * $i ) > $i ).

tff(lameq_f40,axiom,
    ! [A_27a: del,V1m: $i] :
      ( mem(V1m,arr(A_27a,bool))
     => ! [V2x: $i] : ( ap(f40(A_27a,V1m),V2x) = ap(V1m,V2x) ) ) ).

tff(lamtp_f41,type,
    f41: ( del * $i * $i ) > $i ).

tff(lameq_f41,axiom,
    ! [A_27a: del,V1m: $i] :
      ( mem(V1m,arr(A_27a,bool))
     => ! [V3x: $i] :
          ( mem(V3x,A_27a)
         => ! [V4y: $i] : ( ap(f41(A_27a,V1m,V3x),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V1m,V3x)),ap(V1m,V4y))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V4y)) ) ) ) ).

tff(lamtp_f42,type,
    f42: ( del * $i * $i ) > $i ).

tff(lameq_f42,axiom,
    ! [A_27a: del,V1m: $i] :
      ( mem(V1m,arr(A_27a,bool))
     => ! [V0p: $i] :
          ( mem(V0p,arr(A_27a,bool))
         => ! [V3x: $i] : ( ap(f42(A_27a,V1m,V0p),V3x) = ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0p),f41(A_27a,V1m,V3x)) ) ) ) ).

tff(lamtp_f43,type,
    f43: ( del * $i ) > $i ).

tff(lameq_f43,axiom,
    ! [A_27a: del,V0p: $i] :
      ( mem(V0p,arr(A_27a,bool))
     => ! [V1m: $i] : ( ap(f43(A_27a,V0p),V1m) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0p),f40(A_27a,V1m))),ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0p),f42(A_27a,V1m,V0p))) ) ) ).

tff(lamtp_f44,type,
    f44: del > $i ).

tff(lameq_f44,axiom,
    ! [A_27a: del,V0p: $i] : ( ap(f44(A_27a),V0p) = f43(A_27a,V0p) ) ).

tff(ax_thm_2Ebool_2ERES__EXISTS__UNIQUE__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a) = f44(A_27a) ) ).

tff(lamtp_f45,type,
    f45: ( del * $i ) > $i ).

tff(lameq_f45,axiom,
    ! [A_27a: del,V0p: $i] :
      ( mem(V0p,arr(A_27a,bool))
     => ! [V1m: $i] : ( ap(f45(A_27a,V0p),V1m) = ap(c_2Emin_2E_40(A_27a),f37(A_27a,V0p,V1m)) ) ) ).

tff(lamtp_f46,type,
    f46: del > $i ).

tff(lameq_f46,axiom,
    ! [A_27a: del,V0p: $i] : ( ap(f46(A_27a),V0p) = f45(A_27a,V0p) ) ).

tff(ax_thm_2Ebool_2ERES__SELECT__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2ERES__SELECT(A_27a) = f46(A_27a) ) ).

tff(ax_thm_2Ebool_2EBOUNDED__DEF,axiom,
    c_2Ebool_2EBOUNDED = k(bool,c_2Ebool_2ET) ).

tff(ax_thm_2Ebool_2EDATATYPE__TAG__DEF,axiom,
    ! [A_27a: del] : ( c_2Ebool_2EDATATYPE(A_27a) = k(A_27a,c_2Ebool_2ET) ) ).

tff(conj_thm_2Ebool_2ETRUTH,axiom,
    $true ).

tff(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
       => p(inj__o(V1t2)) )
     => ( ( p(inj__o(V1t2))
         => p(inj__o(V0t1)) )
       => ( p(inj__o(V0t1))
        <=> p(inj__o(V1t2)) ) ) ) ).

tff(conj_thm_2Ebool_2EFALSITY,axiom,
    ! [V0t: tp__o] :
      ( $false
     => p(inj__o(V0t)) ) ).

tff(lamtp_f47,type,
    f47: ( del * del * $i ) > $i ).

tff(lameq_f47,axiom,
    ! [A_27b: del,A_27a: del,V0M: $i] :
      ( mem(V0M,arr(A_27a,A_27b))
     => ! [V1x: $i] : ( ap(f47(A_27b,A_27a,V0M),V1x) = ap(V0M,V1x) ) ) ).

tff(conj_thm_2Ebool_2EETA__THM,axiom,
    ! [A_27a: del,A_27b: del,V0M: $i] :
      ( mem(V0M,arr(A_27a,A_27b))
     => ( f47(A_27b,A_27a,V0M) = V0M ) ) ).

tff(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
    ! [V0t: tp__o] :
      ( p(inj__o(V0t))
      | ~ p(inj__o(V0t)) ) ).

tff(lamtp_f48,type,
    f48: ( del * del * $i ) > $i ).

tff(lameq_f48,axiom,
    ! [A_27b: del,A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V2x: $i] : ( ap(f48(A_27b,A_27a,V0f),V2x) = ap(V0f,V2x) ) ) ).

tff(conj_thm_2Ebool_2EBETA__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1y: $i] :
          ( mem(V1y,A_27a)
         => ( ap(f48(A_27b,A_27a,V0f),V1y) = ap(V0f,V1y) ) ) ) ).

tff(conj_thm_2Ebool_2ELET__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1x: $i] :
          ( mem(V1x,A_27a)
         => ( ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V0f),V1x) = ap(V0f,V1x) ) ) ) ).

tff(conj_thm_2Ebool_2EFORALL__THM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,bool))
     => ( p(ap(c_2Ebool_2E_21(A_27a),V0f))
      <=> ! [V1x: $i] :
            ( mem(V1x,A_27a)
           => p(ap(V0f,V1x)) ) ) ) ).

tff(conj_thm_2Ebool_2EEXISTS__THM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,bool))
     => ( p(ap(c_2Ebool_2E_3F(A_27a),V0f))
      <=> ? [V1x: $i] :
            ( mem(V1x,A_27a)
            & p(ap(V0f,V1x)) ) ) ) ).

tff(conj_thm_2Ebool_2EABS__SIMP,axiom,
    ! [A_27a: del,A_27b: del,V0t1: $i] :
      ( mem(V0t1,A_27a)
     => ! [V1t2: $i] :
          ( mem(V1t2,A_27b)
         => ( ap(k(A_27b,V0t1),V1t2) = V0t1 ) ) ) ).

tff(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
    ! [A_27a: del,V0t: tp__o] :
      ( ! [V1x: $i] :
          ( mem(V1x,A_27a)
         => p(inj__o(V0t)) )
    <=> p(inj__o(V0t)) ) ).

tff(conj_thm_2Ebool_2EEXISTS__SIMP,axiom,
    ! [A_27a: del,V0t: tp__o] :
      ( ? [V1x: $i] :
          ( mem(V1x,A_27a)
          & p(inj__o(V0t)) )
    <=> p(inj__o(V0t)) ) ).

tff(conj_thm_2Ebool_2EAND__INTRO__THM,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( p(inj__o(V0t1))
     => ( p(inj__o(V1t2))
       => ( p(inj__o(V0t1))
          & p(inj__o(V1t2)) ) ) ) ).

tff(conj_thm_2Ebool_2EAND1__THM,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
        & p(inj__o(V1t2)) )
     => p(inj__o(V0t1)) ) ).

tff(conj_thm_2Ebool_2EAND2__THM,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
        & p(inj__o(V1t2)) )
     => p(inj__o(V1t2)) ) ).

tff(conj_thm_2Ebool_2ECONJ__SYM,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
        & p(inj__o(V1t2)) )
    <=> ( p(inj__o(V1t2))
        & p(inj__o(V0t1)) ) ) ).

tff(conj_thm_2Ebool_2ECONJ__COMM,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
        & p(inj__o(V1t2)) )
    <=> ( p(inj__o(V1t2))
        & p(inj__o(V0t1)) ) ) ).

tff(conj_thm_2Ebool_2ECONJ__ASSOC,axiom,
    ! [V0t1: tp__o,V1t2: tp__o,V2t3: tp__o] :
      ( ( p(inj__o(V0t1))
        & p(inj__o(V1t2))
        & p(inj__o(V2t3)) )
    <=> ( p(inj__o(V0t1))
        & p(inj__o(V1t2))
        & p(inj__o(V2t3)) ) ) ).

tff(conj_thm_2Ebool_2EOR__INTRO__THM1,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( p(inj__o(V0t1))
     => ( p(inj__o(V0t1))
        | p(inj__o(V1t2)) ) ) ).

tff(conj_thm_2Ebool_2EOR__INTRO__THM2,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( p(inj__o(V1t2))
     => ( p(inj__o(V0t1))
        | p(inj__o(V1t2)) ) ) ).

tff(conj_thm_2Ebool_2EOR__ELIM__THM,axiom,
    ! [V0t: tp__o,V1t1: tp__o,V2t2: tp__o] :
      ( ( p(inj__o(V1t1))
        | p(inj__o(V2t2)) )
     => ( ( p(inj__o(V1t1))
         => p(inj__o(V0t)) )
       => ( ( p(inj__o(V2t2))
           => p(inj__o(V0t)) )
         => p(inj__o(V0t)) ) ) ) ).

tff(conj_thm_2Ebool_2EIMP__F,axiom,
    ! [V0t: tp__o] :
      ( ( p(inj__o(V0t))
       => $false )
     => ~ p(inj__o(V0t)) ) ).

tff(conj_thm_2Ebool_2EF__IMP,axiom,
    ! [V0t: tp__o] :
      ( ~ p(inj__o(V0t))
     => ( p(inj__o(V0t))
       => $false ) ) ).

tff(conj_thm_2Ebool_2ENOT__F,axiom,
    ! [V0t: tp__o] :
      ( ~ p(inj__o(V0t))
     => ( p(inj__o(V0t))
      <=> $false ) ) ).

tff(conj_thm_2Ebool_2ENOT__AND,axiom,
    ! [V0t: tp__o] :
      ~ ( p(inj__o(V0t))
        & ~ p(inj__o(V0t)) ) ).

tff(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
    ! [V0t: tp__o] :
      ( ( ( $true
          & p(inj__o(V0t)) )
      <=> p(inj__o(V0t)) )
      & ( ( p(inj__o(V0t))
          & $true )
      <=> p(inj__o(V0t)) )
      & ( ( $false
          & p(inj__o(V0t)) )
      <=> $false )
      & ( ( p(inj__o(V0t))
          & $false )
      <=> $false )
      & ( ( p(inj__o(V0t))
          & p(inj__o(V0t)) )
      <=> p(inj__o(V0t)) ) ) ).

tff(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
    ! [V0t: tp__o] :
      ( ( ( $true
          | p(inj__o(V0t)) )
      <=> $true )
      & ( ( p(inj__o(V0t))
          | $true )
      <=> $true )
      & ( ( $false
          | p(inj__o(V0t)) )
      <=> p(inj__o(V0t)) )
      & ( ( p(inj__o(V0t))
          | $false )
      <=> p(inj__o(V0t)) )
      & ( ( p(inj__o(V0t))
          | p(inj__o(V0t)) )
      <=> p(inj__o(V0t)) ) ) ).

tff(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
    ! [V0t: tp__o] :
      ( ( ( $true
         => p(inj__o(V0t)) )
      <=> p(inj__o(V0t)) )
      & ( ( p(inj__o(V0t))
         => $true )
      <=> $true )
      & ( ( $false
         => p(inj__o(V0t)) )
      <=> $true )
      & ( ( p(inj__o(V0t))
         => p(inj__o(V0t)) )
      <=> $true )
      & ( ( p(inj__o(V0t))
         => $false )
      <=> ~ p(inj__o(V0t)) ) ) ).

tff(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
    ( ! [V0t: tp__o] :
        ( ~ ~ p(inj__o(V0t))
      <=> p(inj__o(V0t)) )
    & ( ~ $true
    <=> $false )
    & ( ~ $false
    <=> $true ) ) ).

tff(conj_thm_2Ebool_2EEQ__REFL,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ( V0x = V0x ) ) ).

tff(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ( ( V0x = V0x )
      <=> $true ) ) ).

tff(conj_thm_2Ebool_2EEQ__SYM,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ! [V1y: $i] :
          ( mem(V1y,A_27a)
         => ( ( V0x = V1y )
           => ( V1y = V0x ) ) ) ) ).

tff(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ! [V1y: $i] :
          ( mem(V1y,A_27a)
         => ( ( V0x = V1y )
          <=> ( V1y = V0x ) ) ) ) ).

tff(conj_thm_2Ebool_2EEQ__EXT,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1g: $i] :
          ( mem(V1g,arr(A_27a,A_27b))
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( ap(V0f,V2x) = ap(V1g,V2x) ) )
           => ( V0f = V1g ) ) ) ) ).

tff(conj_thm_2Ebool_2EFUN__EQ__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1g: $i] :
          ( mem(V1g,arr(A_27a,A_27b))
         => ( ( V0f = V1g )
          <=> ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( ap(V0f,V2x) = ap(V1g,V2x) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EEQ__TRANS,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ! [V1y: $i] :
          ( mem(V1y,A_27a)
         => ! [V2z: $i] :
              ( mem(V2z,A_27a)
             => ( ( ( V0x = V1y )
                  & ( V1y = V2z ) )
               => ( V0x = V2z ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EBOOL__EQ__DISTINCT,axiom,
    ( ~ ( $true
      <=> $false )
    & ~ ( $false
      <=> $true ) ) ).

tff(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
    ! [V0t: tp__o] :
      ( ( ( $true
        <=> p(inj__o(V0t)) )
      <=> p(inj__o(V0t)) )
      & ( ( p(inj__o(V0t))
        <=> $true )
      <=> p(inj__o(V0t)) )
      & ( ( $false
        <=> p(inj__o(V0t)) )
      <=> ~ p(inj__o(V0t)) )
      & ( ( p(inj__o(V0t))
        <=> $false )
      <=> ~ p(inj__o(V0t)) ) ) ).

tff(conj_thm_2Ebool_2ECOND__CLAUSES,axiom,
    ! [A_27a: del,V0t1: $i] :
      ( mem(V0t1,A_27a)
     => ! [V1t2: $i] :
          ( mem(V1t2,A_27a)
         => ( ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2ET)),V0t1),V1t2) = V0t1 )
            & ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2EF)),V0t1),V1t2) = V1t2 ) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__ID,axiom,
    ! [A_27a: del,V0b: tp__o,V1t: $i] :
      ( mem(V1t,A_27a)
     => ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0b)),V1t),V1t) = V1t ) ) ).

tff(lamtp_f49,type,
    f49: ( del * $i ) > $i ).

tff(lameq_f49,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1x: $i] : ( ap(f49(A_27a,V0P),V1x) = ap(V0P,V1x) ) ) ).

tff(conj_thm_2Ebool_2ESELECT__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( p(ap(V0P,ap(c_2Emin_2E_40(A_27a),f49(A_27a,V0P))))
      <=> ? [V2x: $i] :
            ( mem(V2x,A_27a)
            & p(ap(V0P,V2x)) ) ) ) ).

tff(lamtp_f50,type,
    f50: ( del * $i ) > $i ).

tff(lameq_f50,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ! [V1y: $i] : ( ap(f50(A_27a,V0x),V1y) = ap(ap(c_2Emin_2E_3D(A_27a),V1y),V0x) ) ) ).

tff(conj_thm_2Ebool_2ESELECT__REFL,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ( ap(c_2Emin_2E_40(A_27a),f50(A_27a,V0x)) = V0x ) ) ).

tff(lamtp_f51,type,
    f51: ( del * $i ) > $i ).

tff(lameq_f51,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ! [V1y: $i] : ( ap(f51(A_27a,V0x),V1y) = ap(ap(c_2Emin_2E_3D(A_27a),V0x),V1y) ) ) ).

tff(conj_thm_2Ebool_2ESELECT__REFL__2,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ( ap(c_2Emin_2E_40(A_27a),f51(A_27a,V0x)) = V0x ) ) ).

tff(conj_thm_2Ebool_2ESELECT__UNIQUE,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1x: $i] :
          ( mem(V1x,A_27a)
         => ( ! [V2y: $i] :
                ( mem(V2y,A_27a)
               => ( p(ap(V0P,V2y))
                <=> ( V2y = V1x ) ) )
           => ( ap(c_2Emin_2E_40(A_27a),V0P) = V1x ) ) ) ) ).

tff(conj_thm_2Ebool_2ESELECT__ELIM__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ( ( ? [V2x: $i] :
                  ( mem(V2x,A_27a)
                  & p(ap(V0P,V2x)) )
              & ! [V3x: $i] :
                  ( mem(V3x,A_27a)
                 => ( p(ap(V0P,V3x))
                   => p(ap(V1Q,V3x)) ) ) )
           => p(ap(V1Q,ap(c_2Emin_2E_40(A_27a),V0P))) ) ) ) ).

tff(conj_thm_2Ebool_2ENOT__FORALL__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( ~ ! [V1x: $i] :
              ( mem(V1x,A_27a)
             => p(ap(V0P,V1x)) )
      <=> ? [V2x: $i] :
            ( mem(V2x,A_27a)
            & ~ p(ap(V0P,V2x)) ) ) ) ).

tff(conj_thm_2Ebool_2ENOT__EXISTS__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( ~ ? [V1x: $i] :
              ( mem(V1x,A_27a)
              & p(ap(V0P,V1x)) )
      <=> ! [V2x: $i] :
            ( mem(V2x,A_27a)
           => ~ p(ap(V0P,V2x)) ) ) ) ).

tff(conj_thm_2Ebool_2EFORALL__AND__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( p(ap(V0P,V2x))
                  & p(ap(V1Q,V2x)) ) )
          <=> ( ! [V3x: $i] :
                  ( mem(V3x,A_27a)
                 => p(ap(V0P,V3x)) )
              & ! [V4x: $i] :
                  ( mem(V4x,A_27a)
                 => p(ap(V1Q,V4x)) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => p(ap(V0P,V2x)) )
            & p(inj__o(V1Q)) )
        <=> ! [V3x: $i] :
              ( mem(V3x,A_27a)
             => ( p(ap(V0P,V3x))
                & p(inj__o(V1Q)) ) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ( ( p(inj__o(V0P))
          & ! [V2x: $i] :
              ( mem(V2x,A_27a)
             => p(ap(V1Q,V2x)) ) )
      <=> ! [V3x: $i] :
            ( mem(V3x,A_27a)
           => ( p(inj__o(V0P))
              & p(ap(V1Q,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EEXISTS__OR__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ( ? [V2x: $i] :
                ( mem(V2x,A_27a)
                & ( p(ap(V0P,V2x))
                  | p(ap(V1Q,V2x)) ) )
          <=> ( ? [V3x: $i] :
                  ( mem(V3x,A_27a)
                  & p(ap(V0P,V3x)) )
              | ? [V4x: $i] :
                  ( mem(V4x,A_27a)
                  & p(ap(V1Q,V4x)) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ( ? [V2x: $i] :
                ( mem(V2x,A_27a)
                & p(ap(V0P,V2x)) )
            | p(inj__o(V1Q)) )
        <=> ? [V3x: $i] :
              ( mem(V3x,A_27a)
              & ( p(ap(V0P,V3x))
                | p(inj__o(V1Q)) ) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ( ( p(inj__o(V0P))
          | ? [V2x: $i] :
              ( mem(V2x,A_27a)
              & p(ap(V1Q,V2x)) ) )
      <=> ? [V3x: $i] :
            ( mem(V3x,A_27a)
            & ( p(inj__o(V0P))
              | p(ap(V1Q,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EBOTH__EXISTS__AND__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
      ( ? [V2x: $i] :
          ( mem(V2x,A_27a)
          & p(inj__o(V0P))
          & p(inj__o(V1Q)) )
    <=> ( ? [V3x: $i] :
            ( mem(V3x,A_27a)
            & p(inj__o(V0P)) )
        & ? [V4x: $i] :
            ( mem(V4x,A_27a)
            & p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ? [V2x: $i] :
              ( mem(V2x,A_27a)
              & p(ap(V0P,V2x))
              & p(inj__o(V1Q)) )
        <=> ( ? [V3x: $i] :
                ( mem(V3x,A_27a)
                & p(ap(V0P,V3x)) )
            & p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ( ? [V2x: $i] :
            ( mem(V2x,A_27a)
            & p(inj__o(V0P))
            & p(ap(V1Q,V2x)) )
      <=> ( p(inj__o(V0P))
          & ? [V3x: $i] :
              ( mem(V3x,A_27a)
              & p(ap(V1Q,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EBOTH__FORALL__OR__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
      ( ! [V2x: $i] :
          ( mem(V2x,A_27a)
         => ( p(inj__o(V0P))
            | p(inj__o(V1Q)) ) )
    <=> ( ! [V3x: $i] :
            ( mem(V3x,A_27a)
           => p(inj__o(V0P)) )
        | ! [V4x: $i] :
            ( mem(V4x,A_27a)
           => p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
    ! [A_27a: del,V0Q: tp__o,V1P: $i] :
      ( mem(V1P,arr(A_27a,bool))
     => ( ! [V2x: $i] :
            ( mem(V2x,A_27a)
           => ( p(ap(V1P,V2x))
              | p(inj__o(V0Q)) ) )
      <=> ( ! [V3x: $i] :
              ( mem(V3x,A_27a)
             => p(ap(V1P,V3x)) )
          | p(inj__o(V0Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ( ! [V2x: $i] :
            ( mem(V2x,A_27a)
           => ( p(inj__o(V0P))
              | p(ap(V1Q,V2x)) ) )
      <=> ( p(inj__o(V0P))
          | ! [V3x: $i] :
              ( mem(V3x,A_27a)
             => p(ap(V1Q,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EBOTH__FORALL__IMP__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
      ( ! [V2x: $i] :
          ( mem(V2x,A_27a)
         => ( p(inj__o(V0P))
           => p(inj__o(V1Q)) ) )
    <=> ( ? [V3x: $i] :
            ( mem(V3x,A_27a)
            & p(inj__o(V0P)) )
       => ! [V4x: $i] :
            ( mem(V4x,A_27a)
           => p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__FORALL__IMP__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ! [V2x: $i] :
              ( mem(V2x,A_27a)
             => ( p(ap(V0P,V2x))
               => p(inj__o(V1Q)) ) )
        <=> ( ? [V3x: $i] :
                ( mem(V3x,A_27a)
                & p(ap(V0P,V3x)) )
           => p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__FORALL__IMP__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ( ! [V2x: $i] :
            ( mem(V2x,A_27a)
           => ( p(inj__o(V0P))
             => p(ap(V1Q,V2x)) ) )
      <=> ( p(inj__o(V0P))
         => ! [V3x: $i] :
              ( mem(V3x,A_27a)
             => p(ap(V1Q,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EBOTH__EXISTS__IMP__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
      ( ? [V2x: $i] :
          ( mem(V2x,A_27a)
          & ( p(inj__o(V0P))
           => p(inj__o(V1Q)) ) )
    <=> ( ! [V3x: $i] :
            ( mem(V3x,A_27a)
           => p(inj__o(V0P)) )
       => ? [V4x: $i] :
            ( mem(V4x,A_27a)
            & p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__EXISTS__IMP__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ? [V2x: $i] :
              ( mem(V2x,A_27a)
              & ( p(ap(V0P,V2x))
               => p(inj__o(V1Q)) ) )
        <=> ( ! [V3x: $i] :
                ( mem(V3x,A_27a)
               => p(ap(V0P,V3x)) )
           => p(inj__o(V1Q)) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__EXISTS__IMP__THM,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ( ? [V2x: $i] :
            ( mem(V2x,A_27a)
            & ( p(inj__o(V0P))
             => p(ap(V1Q,V2x)) ) )
      <=> ( p(inj__o(V0P))
         => ? [V3x: $i] :
              ( mem(V3x,A_27a)
              & p(ap(V1Q,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EOR__IMP__THM,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ( p(inj__o(V0A))
      <=> ( p(inj__o(V1B))
          | p(inj__o(V0A)) ) )
    <=> ( p(inj__o(V1B))
       => p(inj__o(V0A)) ) ) ).

tff(conj_thm_2Ebool_2ENOT__IMP,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ~ ( p(inj__o(V0A))
         => p(inj__o(V1B)) )
    <=> ( p(inj__o(V0A))
        & ~ p(inj__o(V1B)) ) ) ).

tff(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
    ! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
      ( ( p(inj__o(V0A))
        | p(inj__o(V1B))
        | p(inj__o(V2C)) )
    <=> ( p(inj__o(V0A))
        | p(inj__o(V1B))
        | p(inj__o(V2C)) ) ) ).

tff(conj_thm_2Ebool_2EDISJ__SYM,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ( p(inj__o(V0A))
        | p(inj__o(V1B)) )
    <=> ( p(inj__o(V1B))
        | p(inj__o(V0A)) ) ) ).

tff(conj_thm_2Ebool_2EDISJ__COMM,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ( p(inj__o(V0A))
        | p(inj__o(V1B)) )
    <=> ( p(inj__o(V1B))
        | p(inj__o(V0A)) ) ) ).

tff(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ( ~ ( p(inj__o(V0A))
            & p(inj__o(V1B)) )
      <=> ( ~ p(inj__o(V0A))
          | ~ p(inj__o(V1B)) ) )
      & ( ~ ( p(inj__o(V0A))
            | p(inj__o(V1B)) )
      <=> ( ~ p(inj__o(V0A))
          & ~ p(inj__o(V1B)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__AND__OVER__OR,axiom,
    ! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
      ( ( p(inj__o(V0A))
        & ( p(inj__o(V1B))
          | p(inj__o(V2C)) ) )
    <=> ( ( p(inj__o(V0A))
          & p(inj__o(V1B)) )
        | ( p(inj__o(V0A))
          & p(inj__o(V2C)) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__AND__OVER__OR,axiom,
    ! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
      ( ( ( p(inj__o(V1B))
          | p(inj__o(V2C)) )
        & p(inj__o(V0A)) )
    <=> ( ( p(inj__o(V1B))
          & p(inj__o(V0A)) )
        | ( p(inj__o(V2C))
          & p(inj__o(V0A)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__OR__OVER__AND,axiom,
    ! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
      ( ( p(inj__o(V0A))
        | ( p(inj__o(V1B))
          & p(inj__o(V2C)) ) )
    <=> ( ( p(inj__o(V0A))
          | p(inj__o(V1B)) )
        & ( p(inj__o(V0A))
          | p(inj__o(V2C)) ) ) ) ).

tff(conj_thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
    ! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
      ( ( ( p(inj__o(V1B))
          & p(inj__o(V2C)) )
        | p(inj__o(V0A)) )
    <=> ( ( p(inj__o(V1B))
          | p(inj__o(V0A)) )
        & ( p(inj__o(V2C))
          | p(inj__o(V0A)) ) ) ) ).

tff(conj_thm_2Ebool_2EIMP__DISJ__THM,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ( p(inj__o(V0A))
       => p(inj__o(V1B)) )
    <=> ( ~ p(inj__o(V0A))
        | p(inj__o(V1B)) ) ) ).

tff(conj_thm_2Ebool_2EDISJ__IMP__THM,axiom,
    ! [V0P: tp__o,V1Q: tp__o,V2R: tp__o] :
      ( ( ( p(inj__o(V0P))
          | p(inj__o(V1Q)) )
       => p(inj__o(V2R)) )
    <=> ( ( p(inj__o(V0P))
         => p(inj__o(V2R)) )
        & ( p(inj__o(V1Q))
         => p(inj__o(V2R)) ) ) ) ).

tff(conj_thm_2Ebool_2EIMP__CONJ__THM,axiom,
    ! [V0P: tp__o,V1Q: tp__o,V2R: tp__o] :
      ( ( p(inj__o(V0P))
       => ( p(inj__o(V1Q))
          & p(inj__o(V2R)) ) )
    <=> ( ( p(inj__o(V0P))
         => p(inj__o(V1Q)) )
        & ( p(inj__o(V0P))
         => p(inj__o(V2R)) ) ) ) ).

tff(conj_thm_2Ebool_2EIMP__F__EQ__F,axiom,
    ! [V0t: tp__o] :
      ( ( p(inj__o(V0t))
       => $false )
    <=> ( p(inj__o(V0t))
      <=> $false ) ) ).

tff(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
    ! [V0t1: tp__o,V1t2: tp__o,V2t3: tp__o] :
      ( ( p(inj__o(V0t1))
       => ( p(inj__o(V1t2))
         => p(inj__o(V2t3)) ) )
    <=> ( ( p(inj__o(V0t1))
          & p(inj__o(V1t2)) )
       => p(inj__o(V2t3)) ) ) ).

tff(conj_thm_2Ebool_2EEQ__IMP__THM,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
      <=> p(inj__o(V1t2)) )
    <=> ( ( p(inj__o(V0t1))
         => p(inj__o(V1t2)) )
        & ( p(inj__o(V1t2))
         => p(inj__o(V0t1)) ) ) ) ).

tff(conj_thm_2Ebool_2EEQ__EXPAND,axiom,
    ! [V0t1: tp__o,V1t2: tp__o] :
      ( ( p(inj__o(V0t1))
      <=> p(inj__o(V1t2)) )
    <=> ( ( p(inj__o(V0t1))
          & p(inj__o(V1t2)) )
        | ( ~ p(inj__o(V0t1))
          & ~ p(inj__o(V1t2)) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__RATOR,axiom,
    ! [A_27a: del,A_27b: del,V0b: tp__o,V1f: $i] :
      ( mem(V1f,arr(A_27a,A_27b))
     => ! [V2g: $i] :
          ( mem(V2g,arr(A_27a,A_27b))
         => ! [V3x: $i] :
              ( mem(V3x,A_27a)
             => ( ap(ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,A_27b)),inj__o(V0b)),V1f),V2g),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),inj__o(V0b)),ap(V1f,V3x)),ap(V2g,V3x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__RAND,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1b: tp__o,V2x: $i] :
          ( mem(V2x,A_27a)
         => ! [V3y: $i] :
              ( mem(V3y,A_27a)
             => ( ap(V0f,ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1b)),V2x),V3y)) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),inj__o(V1b)),ap(V0f,V2x)),ap(V0f,V3y)) ) ) ) ) ).

tff(lamtp_f52,type,
    f52: ( del * del * tp__o * $i * $i ) > $i ).

tff(lameq_f52,axiom,
    ! [A_27b: del,A_27a: del,V0b: tp__o,V1f: $i] :
      ( mem(V1f,arr(A_27a,A_27b))
     => ! [V2g: $i] :
          ( mem(V2g,arr(A_27a,A_27b))
         => ! [V3x: $i] : ( ap(f52(A_27b,A_27a,V0b,V1f,V2g),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),inj__o(V0b)),ap(V1f,V3x)),ap(V2g,V3x)) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__ABS,axiom,
    ! [A_27a: del,A_27b: del,V0b: tp__o,V1f: $i] :
      ( mem(V1f,arr(A_27a,A_27b))
     => ! [V2g: $i] :
          ( mem(V2g,arr(A_27a,A_27b))
         => ( f52(A_27b,A_27a,V0b,V1f,V2g) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,A_27b)),inj__o(V0b)),V1f),V2g) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__EXPAND,axiom,
    ! [V0b: tp__o,V1t1: tp__o,V2t2: tp__o] :
      ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0b)),inj__o(V1t1)),inj__o(V2t2)))
    <=> ( ( ~ p(inj__o(V0b))
          | p(inj__o(V1t1)) )
        & ( p(inj__o(V0b))
          | p(inj__o(V2t2)) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__EXPAND__IMP,axiom,
    ! [V0b: tp__o,V1t1: tp__o,V2t2: tp__o] :
      ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0b)),inj__o(V1t1)),inj__o(V2t2)))
    <=> ( ( p(inj__o(V0b))
         => p(inj__o(V1t1)) )
        & ( ~ p(inj__o(V0b))
         => p(inj__o(V2t2)) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__EXPAND__OR,axiom,
    ! [V0b: tp__o,V1t1: tp__o,V2t2: tp__o] :
      ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0b)),inj__o(V1t1)),inj__o(V2t2)))
    <=> ( ( p(inj__o(V0b))
          & p(inj__o(V1t1)) )
        | ( ~ p(inj__o(V0b))
          & p(inj__o(V2t2)) ) ) ) ).

tff(conj_thm_2Ebool_2ETYPE__DEFINITION__THM,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1rep: $i] :
          ( mem(V1rep,arr(A_27b,A_27a))
         => ( p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),V0P),V1rep))
          <=> ( ! [V2x_27: $i] :
                  ( mem(V2x_27,A_27b)
                 => ! [V3x_27_27: $i] :
                      ( mem(V3x_27_27,A_27b)
                     => ( ( ap(V1rep,V2x_27) = ap(V1rep,V3x_27_27) )
                       => ( V2x_27 = V3x_27_27 ) ) ) )
              & ! [V4x: $i] :
                  ( mem(V4x,A_27a)
                 => ( p(ap(V0P,V4x))
                  <=> ? [V5x_27: $i] :
                        ( mem(V5x_27,A_27b)
                        & ( V4x = ap(V1rep,V5x_27) ) ) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EONTO__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ( p(ap(c_2Ebool_2EONTO(A_27a,A_27b),V0f))
      <=> ! [V1y: $i] :
            ( mem(V1y,A_27b)
           => ? [V2x: $i] :
                ( mem(V2x,A_27a)
                & ( V1y = ap(V0f,V2x) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EONE__ONE__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ( p(ap(c_2Ebool_2EONE__ONE(A_27a,A_27b),V0f))
      <=> ! [V1x1: $i] :
            ( mem(V1x1,A_27a)
           => ! [V2x2: $i] :
                ( mem(V2x2,A_27a)
               => ( ( ap(V0f,V1x1) = ap(V0f,V2x2) )
                 => ( V1x1 = V2x2 ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EABS__REP__THM,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( ? [V1rep: $i] :
            ( mem(V1rep,arr(A_27b,A_27a))
            & p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),V0P),V1rep)) )
       => ? [V2rep: $i] :
            ( mem(V2rep,arr(A_27b,A_27a))
            & ? [V3abs: $i] :
                ( mem(V3abs,arr(A_27a,A_27b))
                & ! [V4a: $i] :
                    ( mem(V4a,A_27b)
                   => ( ap(V3abs,ap(V2rep,V4a)) = V4a ) )
                & ! [V5r: $i] :
                    ( mem(V5r,A_27a)
                   => ( p(ap(V0P,V5r))
                    <=> ( ap(V2rep,ap(V3abs,V5r)) = V5r ) ) ) ) ) ) ) ).

tff(lamtp_f53,type,
    f53: ( del * del * $i ) > $i ).

tff(lameq_f53,axiom,
    ! [A_27b: del,A_27a: del,V1N: $i] :
      ( mem(V1N,arr(A_27a,A_27b))
     => ! [V3x: $i] : ( ap(f53(A_27b,A_27a,V1N),V3x) = ap(V1N,V3x) ) ) ).

tff(lamtp_f54,type,
    f54: ( del * del * $i * $i ) > $i ).

tff(lameq_f54,axiom,
    ! [A_27b: del,A_27a: del,V1N: $i] :
      ( mem(V1N,arr(A_27a,A_27b))
     => ! [V0P: $i] :
          ( mem(V0P,arr(A_27b,bool))
         => ! [V4x: $i] : ( ap(f54(A_27b,A_27a,V1N,V0P),V4x) = ap(V0P,ap(V1N,V4x)) ) ) ) ).

tff(conj_thm_2Ebool_2ELET__RAND,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(A_27b,bool))
     => ! [V1N: $i] :
          ( mem(V1N,arr(A_27a,A_27b))
         => ! [V2M: $i] :
              ( mem(V2M,A_27a)
             => ( p(ap(V0P,ap(ap(c_2Ebool_2ELET(A_27a,A_27b),f53(A_27b,A_27a,V1N)),V2M)))
              <=> p(ap(ap(c_2Ebool_2ELET(A_27a,bool),f54(A_27b,A_27a,V1N,V0P)),V2M)) ) ) ) ) ).

tff(lamtp_f55,type,
    f55: ( del * del * del * $i ) > $i ).

tff(lameq_f55,axiom,
    ! [A_27c: del,A_27b: del,A_27a: del,V0N: $i] :
      ( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
     => ! [V3x: $i] : ( ap(f55(A_27c,A_27b,A_27a,V0N),V3x) = ap(V0N,V3x) ) ) ).

tff(lamtp_f56,type,
    f56: ( del * del * del * $i * $i ) > $i ).

tff(lameq_f56,axiom,
    ! [A_27b: del,A_27c: del,A_27a: del,V0N: $i] :
      ( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
     => ! [V2b: $i] :
          ( mem(V2b,A_27b)
         => ! [V4x: $i] : ( ap(f56(A_27b,A_27c,A_27a,V0N,V2b),V4x) = ap(ap(V0N,V4x),V2b) ) ) ) ).

tff(conj_thm_2Ebool_2ELET__RATOR,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0N: $i] :
      ( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
     => ! [V1M: $i] :
          ( mem(V1M,A_27a)
         => ! [V2b: $i] :
              ( mem(V2b,A_27b)
             => ( ap(ap(ap(c_2Ebool_2ELET(A_27a,arr(A_27b,A_27c)),f55(A_27c,A_27b,A_27a,V0N)),V1M),V2b) = ap(ap(c_2Ebool_2ELET(A_27a,A_27c),f56(A_27b,A_27c,A_27a,V0N,V2b)),V1M) ) ) ) ) ).

tff(conj_thm_2Ebool_2ESWAP__FORALL__THM,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,arr(A_27b,bool)))
     => ( ! [V1x: $i] :
            ( mem(V1x,A_27a)
           => ! [V2y: $i] :
                ( mem(V2y,A_27b)
               => p(ap(ap(V0P,V1x),V2y)) ) )
      <=> ! [V3y: $i] :
            ( mem(V3y,A_27b)
           => ! [V4x: $i] :
                ( mem(V4x,A_27a)
               => p(ap(ap(V0P,V4x),V3y)) ) ) ) ) ).

tff(conj_thm_2Ebool_2ESWAP__EXISTS__THM,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,arr(A_27b,bool)))
     => ( ? [V1x: $i] :
            ( mem(V1x,A_27a)
            & ? [V2y: $i] :
                ( mem(V2y,A_27b)
                & p(ap(ap(V0P,V1x),V2y)) ) )
      <=> ? [V3y: $i] :
            ( mem(V3y,A_27b)
            & ? [V4x: $i] :
                ( mem(V4x,A_27a)
                & p(ap(ap(V0P,V4x),V3y)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EEXISTS__UNIQUE__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( p(ap(c_2Ebool_2E_3F_21(A_27a),f49(A_27a,V0P)))
      <=> ( ? [V2x: $i] :
              ( mem(V2x,A_27a)
              & p(ap(V0P,V2x)) )
          & ! [V3x: $i] :
              ( mem(V3x,A_27a)
             => ! [V4y: $i] :
                  ( mem(V4y,A_27a)
                 => ( ( p(ap(V0P,V3x))
                      & p(ap(V0P,V4y)) )
                   => ( V3x = V4y ) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ELET__CONG,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1g: $i] :
          ( mem(V1g,arr(A_27a,A_27b))
         => ! [V2M: $i] :
              ( mem(V2M,A_27a)
             => ! [V3N: $i] :
                  ( mem(V3N,A_27a)
                 => ( ( ( V2M = V3N )
                      & ! [V4x: $i] :
                          ( mem(V4x,A_27a)
                         => ( ( V4x = V3N )
                           => ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) )
                   => ( ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V0f),V2M) = ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V1g),V3N) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EIMP__CONG,axiom,
    ! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
      ( ( ( p(inj__o(V0x))
        <=> p(inj__o(V1x_27)) )
        & ( p(inj__o(V1x_27))
         => ( p(inj__o(V2y))
          <=> p(inj__o(V3y_27)) ) ) )
     => ( ( p(inj__o(V0x))
         => p(inj__o(V2y)) )
      <=> ( p(inj__o(V1x_27))
         => p(inj__o(V3y_27)) ) ) ) ).

tff(conj_thm_2Ebool_2EAND__CONG,axiom,
    ! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
      ( ( ( p(inj__o(V2Q))
         => ( p(inj__o(V0P))
          <=> p(inj__o(V1P_27)) ) )
        & ( p(inj__o(V1P_27))
         => ( p(inj__o(V2Q))
          <=> p(inj__o(V3Q_27)) ) ) )
     => ( ( p(inj__o(V0P))
          & p(inj__o(V2Q)) )
      <=> ( p(inj__o(V1P_27))
          & p(inj__o(V3Q_27)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__AND__CONG,axiom,
    ! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
      ( ( ( p(inj__o(V0P))
        <=> p(inj__o(V1P_27)) )
        & ( p(inj__o(V1P_27))
         => ( p(inj__o(V2Q))
          <=> p(inj__o(V3Q_27)) ) ) )
     => ( ( p(inj__o(V0P))
          & p(inj__o(V2Q)) )
      <=> ( p(inj__o(V1P_27))
          & p(inj__o(V3Q_27)) ) ) ) ).

tff(conj_thm_2Ebool_2EOR__CONG,axiom,
    ! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
      ( ( ( ~ p(inj__o(V2Q))
         => ( p(inj__o(V0P))
          <=> p(inj__o(V1P_27)) ) )
        & ( ~ p(inj__o(V1P_27))
         => ( p(inj__o(V2Q))
          <=> p(inj__o(V3Q_27)) ) ) )
     => ( ( p(inj__o(V0P))
          | p(inj__o(V2Q)) )
      <=> ( p(inj__o(V1P_27))
          | p(inj__o(V3Q_27)) ) ) ) ).

tff(conj_thm_2Ebool_2ELEFT__OR__CONG,axiom,
    ! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
      ( ( ( p(inj__o(V0P))
        <=> p(inj__o(V1P_27)) )
        & ( ~ p(inj__o(V1P_27))
         => ( p(inj__o(V2Q))
          <=> p(inj__o(V3Q_27)) ) ) )
     => ( ( p(inj__o(V0P))
          | p(inj__o(V2Q)) )
      <=> ( p(inj__o(V1P_27))
          | p(inj__o(V3Q_27)) ) ) ) ).

tff(conj_thm_2Ebool_2ECOND__CONG,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: tp__o,V2x: $i] :
      ( mem(V2x,A_27a)
     => ! [V3x_27: $i] :
          ( mem(V3x_27,A_27a)
         => ! [V4y: $i] :
              ( mem(V4y,A_27a)
             => ! [V5y_27: $i] :
                  ( mem(V5y_27,A_27a)
                 => ( ( ( p(inj__o(V0P))
                      <=> p(inj__o(V1Q)) )
                      & ( p(inj__o(V1Q))
                       => ( V2x = V3x_27 ) )
                      & ( ~ p(inj__o(V1Q))
                       => ( V4y = V5y_27 ) ) )
                   => ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0P)),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1Q)),V3x_27),V5y_27) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__FORALL__CONG,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ! [V2f: $i] :
              ( mem(V2f,arr(A_27a,bool))
             => ! [V3g: $i] :
                  ( mem(V3g,arr(A_27a,bool))
                 => ( ( V0P = V1Q )
                   => ( ! [V4x: $i] :
                          ( mem(V4x,A_27a)
                         => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1Q))
                           => ( p(ap(V2f,V4x))
                            <=> p(ap(V3g,V4x)) ) ) )
                     => ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),V2f))
                      <=> p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V1Q),V3g)) ) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__EXISTS__CONG,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ! [V2f: $i] :
              ( mem(V2f,arr(A_27a,bool))
             => ! [V3g: $i] :
                  ( mem(V3g,arr(A_27a,bool))
                 => ( ( V0P = V1Q )
                   => ( ! [V4x: $i] :
                          ( mem(V4x,A_27a)
                         => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1Q))
                           => ( p(ap(V2f,V4x))
                            <=> p(ap(V3g,V4x)) ) ) )
                     => ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),V2f))
                      <=> p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V1Q),V3g)) ) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EMONO__AND,axiom,
    ! [V0x: tp__o,V1y: tp__o,V2z: tp__o,V3w: tp__o] :
      ( ( ( p(inj__o(V0x))
         => p(inj__o(V1y)) )
        & ( p(inj__o(V2z))
         => p(inj__o(V3w)) ) )
     => ( ( p(inj__o(V0x))
          & p(inj__o(V2z)) )
       => ( p(inj__o(V1y))
          & p(inj__o(V3w)) ) ) ) ).

tff(conj_thm_2Ebool_2EMONO__OR,axiom,
    ! [V0x: tp__o,V1y: tp__o,V2z: tp__o,V3w: tp__o] :
      ( ( ( p(inj__o(V0x))
         => p(inj__o(V1y)) )
        & ( p(inj__o(V2z))
         => p(inj__o(V3w)) ) )
     => ( ( p(inj__o(V0x))
          | p(inj__o(V2z)) )
       => ( p(inj__o(V1y))
          | p(inj__o(V3w)) ) ) ) ).

tff(conj_thm_2Ebool_2EMONO__IMP,axiom,
    ! [V0y: tp__o,V1x: tp__o,V2z: tp__o,V3w: tp__o] :
      ( ( ( p(inj__o(V0y))
         => p(inj__o(V1x)) )
        & ( p(inj__o(V2z))
         => p(inj__o(V3w)) ) )
     => ( ( p(inj__o(V1x))
         => p(inj__o(V2z)) )
       => ( p(inj__o(V0y))
         => p(inj__o(V3w)) ) ) ) ).

tff(conj_thm_2Ebool_2EMONO__NOT,axiom,
    ! [V0y: tp__o,V1x: tp__o] :
      ( ( p(inj__o(V0y))
       => p(inj__o(V1x)) )
     => ( ~ p(inj__o(V1x))
       => ~ p(inj__o(V0y)) ) ) ).

tff(conj_thm_2Ebool_2EMONO__NOT__EQ,axiom,
    ! [V0y: tp__o,V1x: tp__o] :
      ( ( p(inj__o(V0y))
       => p(inj__o(V1x)) )
    <=> ( ~ p(inj__o(V1x))
       => ~ p(inj__o(V0y)) ) ) ).

tff(conj_thm_2Ebool_2EMONO__ALL,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( p(ap(V0P,V2x))
                 => p(ap(V1Q,V2x)) ) )
           => ( ! [V3x: $i] :
                  ( mem(V3x,A_27a)
                 => p(ap(V0P,V3x)) )
             => ! [V4x: $i] :
                  ( mem(V4x,A_27a)
                 => p(ap(V1Q,V4x)) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EMONO__EXISTS,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( p(ap(V0P,V2x))
                 => p(ap(V1Q,V2x)) ) )
           => ( ? [V3x: $i] :
                  ( mem(V3x,A_27a)
                  & p(ap(V0P,V3x)) )
             => ? [V4x: $i] :
                  ( mem(V4x,A_27a)
                  & p(ap(V1Q,V4x)) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EMONO__COND,axiom,
    ! [V0x: tp__o,V1y: tp__o,V2z: tp__o,V3w: tp__o,V4b: tp__o] :
      ( ( p(inj__o(V0x))
       => p(inj__o(V1y)) )
     => ( ( p(inj__o(V2z))
         => p(inj__o(V3w)) )
       => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V4b)),inj__o(V0x)),inj__o(V2z)))
         => p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V4b)),inj__o(V1y)),inj__o(V3w))) ) ) ) ).

tff(conj_thm_2Ebool_2EEXISTS__REFL,axiom,
    ! [A_27a: del,V0a: $i] :
      ( mem(V0a,A_27a)
     => ? [V1x: $i] :
          ( mem(V1x,A_27a)
          & ( V1x = V0a ) ) ) ).

tff(lamtp_f57,type,
    f57: ( del * $i ) > $i ).

tff(lameq_f57,axiom,
    ! [A_27a: del,V0a: $i] :
      ( mem(V0a,A_27a)
     => ! [V1x: $i] : ( ap(f57(A_27a,V0a),V1x) = ap(ap(c_2Emin_2E_3D(A_27a),V1x),V0a) ) ) ).

tff(conj_thm_2Ebool_2EEXISTS__UNIQUE__REFL,axiom,
    ! [A_27a: del,V0a: $i] :
      ( mem(V0a,A_27a)
     => p(ap(c_2Ebool_2E_3F_21(A_27a),f57(A_27a,V0a))) ) ).

tff(conj_thm_2Ebool_2EUNWIND__THM1,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1a: $i] :
          ( mem(V1a,A_27a)
         => ( ? [V2x: $i] :
                ( mem(V2x,A_27a)
                & ( V1a = V2x )
                & p(ap(V0P,V2x)) )
          <=> p(ap(V0P,V1a)) ) ) ) ).

tff(conj_thm_2Ebool_2EUNWIND__THM2,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1a: $i] :
          ( mem(V1a,A_27a)
         => ( ? [V2x: $i] :
                ( mem(V2x,A_27a)
                & ( V2x = V1a )
                & p(ap(V0P,V2x)) )
          <=> p(ap(V0P,V1a)) ) ) ) ).

tff(conj_thm_2Ebool_2EUNWIND__FORALL__THM1,axiom,
    ! [A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,bool))
     => ! [V1v: $i] :
          ( mem(V1v,A_27a)
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( ( V1v = V2x )
                 => p(ap(V0f,V2x)) ) )
          <=> p(ap(V0f,V1v)) ) ) ) ).

tff(conj_thm_2Ebool_2EUNWIND__FORALL__THM2,axiom,
    ! [A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,bool))
     => ! [V1v: $i] :
          ( mem(V1v,A_27a)
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( ( V2x = V1v )
                 => p(ap(V0f,V2x)) ) )
          <=> p(ap(V0f,V1v)) ) ) ) ).

tff(conj_thm_2Ebool_2ESKOLEM__THM,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,arr(A_27b,bool)))
     => ( ! [V1x: $i] :
            ( mem(V1x,A_27a)
           => ? [V2y: $i] :
                ( mem(V2y,A_27b)
                & p(ap(ap(V0P,V1x),V2y)) ) )
      <=> ? [V3f: $i] :
            ( mem(V3f,arr(A_27a,A_27b))
            & ! [V4x: $i] :
                ( mem(V4x,A_27a)
               => p(ap(ap(V0P,V4x),ap(V3f,V4x))) ) ) ) ) ).

tff(conj_thm_2Ebool_2Ebool__case__thm,axiom,
    ! [A_27a: del] :
      ( ! [V0t1: $i] :
          ( mem(V0t1,A_27a)
         => ! [V1t2: $i] :
              ( mem(V1t2,A_27a)
             => ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2ET)),V0t1),V1t2) = V0t1 ) ) )
      & ! [V2t1: $i] :
          ( mem(V2t1,A_27a)
         => ! [V3t2: $i] :
              ( mem(V3t2,A_27a)
             => ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2EF)),V2t1),V3t2) = V3t2 ) ) ) ) ).

tff(conj_thm_2Ebool_2Ebool__case__ID,axiom,
    ! [A_27a: del,V0b: tp__o,V1t: $i] :
      ( mem(V1t,A_27a)
     => ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0b)),V1t),V1t) = V1t ) ) ).

tff(conj_thm_2Ebool_2EboolAxiom,axiom,
    ! [A_27a: del,V0t1: $i] :
      ( mem(V0t1,A_27a)
     => ! [V1t2: $i] :
          ( mem(V1t2,A_27a)
         => ? [V2fn: $i] :
              ( mem(V2fn,arr(bool,A_27a))
              & ( ap(V2fn,inj__o(fo__c_2Ebool_2ET)) = V0t1 )
              & ( ap(V2fn,inj__o(fo__c_2Ebool_2EF)) = V1t2 ) ) ) ) ).

tff(conj_thm_2Ebool_2Ebool__INDUCT,axiom,
    ! [V0P: $i] :
      ( mem(V0P,arr(bool,bool))
     => ( ( p(ap(V0P,inj__o(fo__c_2Ebool_2ET)))
          & p(ap(V0P,inj__o(fo__c_2Ebool_2EF))) )
       => ! [V1b: tp__o] : p(ap(V0P,inj__o(V1b))) ) ) ).

tff(conj_thm_2Ebool_2Ebool__case__CONG,axiom,
    ! [A_27a: del,V0P: tp__o,V1Q: tp__o,V2x: $i] :
      ( mem(V2x,A_27a)
     => ! [V3x_27: $i] :
          ( mem(V3x_27,A_27a)
         => ! [V4y: $i] :
              ( mem(V4y,A_27a)
             => ! [V5y_27: $i] :
                  ( mem(V5y_27,A_27a)
                 => ( ( ( p(inj__o(V0P))
                      <=> p(inj__o(V1Q)) )
                      & ( p(inj__o(V1Q))
                       => ( V2x = V3x_27 ) )
                      & ( ~ p(inj__o(V1Q))
                       => ( V4y = V5y_27 ) ) )
                   => ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0P)),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1Q)),V3x_27),V5y_27) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EFORALL__BOOL,axiom,
    ! [V0P: $i] :
      ( mem(V0P,arr(bool,bool))
     => ( ! [V1b: tp__o] : p(ap(V0P,inj__o(V1b)))
      <=> ( p(ap(V0P,inj__o(fo__c_2Ebool_2ET)))
          & p(ap(V0P,inj__o(fo__c_2Ebool_2EF))) ) ) ) ).

tff(lamtp_f58,type,
    f58: ( del * $i * $i ) > $i ).

tff(lameq_f58,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ! [V2x: $i] : ( ap(f58(A_27a,V0P,V1Q),V2x) = ap(ap(c_2Ebool_2E_5C_2F,ap(V0P,V2x)),ap(V1Q,V2x)) ) ) ) ).

tff(lamtp_f59,type,
    f59: ( del * $i ) > $i ).

tff(lameq_f59,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V3x: $i] : ( ap(f59(A_27a,V0P),V3x) = ap(V0P,V3x) ) ) ).

tff(lamtp_f60,type,
    f60: ( del * $i ) > $i ).

tff(lameq_f60,axiom,
    ! [A_27a: del,V1Q: $i] :
      ( mem(V1Q,arr(A_27a,bool))
     => ! [V4x: $i] : ( ap(f60(A_27a,V1Q),V4x) = ap(V1Q,V4x) ) ) ).

tff(conj_thm_2Ebool_2EUEXISTS__OR__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: $i] :
          ( mem(V1Q,arr(A_27a,bool))
         => ( p(ap(c_2Ebool_2E_3F_21(A_27a),f58(A_27a,V0P,V1Q)))
           => ( p(ap(c_2Ebool_2E_3F_21(A_27a),f59(A_27a,V0P)))
              | p(ap(c_2Ebool_2E_3F_21(A_27a),f60(A_27a,V1Q))) ) ) ) ) ).

tff(conj_thm_2Ebool_2EUEXISTS__SIMP,axiom,
    ! [A_27a: del,V0t: tp__o] :
      ( p(ap(c_2Ebool_2E_3F_21(A_27a),k(A_27a,inj__o(V0t))))
    <=> ( p(inj__o(V0t))
        & ! [V2x: $i] :
            ( mem(V2x,A_27a)
           => ! [V3y: $i] :
                ( mem(V3y,A_27a)
               => ( V2x = V3y ) ) ) ) ) ).

tff(ax_thm_2Ebool_2ERES__ABSTRACT__DEF,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ! [V0p: $i] :
          ( mem(V0p,arr(A_27a,bool))
         => ! [V1m: $i] :
              ( mem(V1m,arr(A_27a,A_27b))
             => ! [V2x: $i] :
                  ( mem(V2x,A_27a)
                 => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p))
                   => ( ap(ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V0p),V1m),V2x) = ap(V1m,V2x) ) ) ) ) )
      & ! [V3p: $i] :
          ( mem(V3p,arr(A_27a,bool))
         => ! [V4m1: $i] :
              ( mem(V4m1,arr(A_27a,A_27b))
             => ! [V5m2: $i] :
                  ( mem(V5m2,arr(A_27a,A_27b))
                 => ( ! [V6x: $i] :
                        ( mem(V6x,A_27a)
                       => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V3p))
                         => ( ap(V4m1,V6x) = ap(V5m2,V6x) ) ) )
                   => ( ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V3p),V4m1) = ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V3p),V5m2) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__FORALL__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,bool))
         => ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),V1f))
          <=> ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
                 => p(ap(V1f,V2x)) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__EXISTS__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,bool))
         => ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),V1f))
          <=> ? [V2x: $i] :
                ( mem(V2x,A_27a)
                & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
                & p(ap(V1f,V2x)) ) ) ) ) ).

tff(lamtp_f61,type,
    f61: ( del * $i ) > $i ).

tff(lameq_f61,axiom,
    ! [A_27a: del,V1f: $i] :
      ( mem(V1f,arr(A_27a,bool))
     => ! [V2x: $i] : ( ap(f61(A_27a,V1f),V2x) = ap(V1f,V2x) ) ) ).

tff(lamtp_f62,type,
    f62: ( del * $i * $i ) > $i ).

tff(lameq_f62,axiom,
    ! [A_27a: del,V1f: $i] :
      ( mem(V1f,arr(A_27a,bool))
     => ! [V3x: $i] :
          ( mem(V3x,A_27a)
         => ! [V4y: $i] : ( ap(f62(A_27a,V1f,V3x),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V1f,V3x)),ap(V1f,V4y))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V4y)) ) ) ) ).

tff(lamtp_f63,type,
    f63: ( del * $i * $i ) > $i ).

tff(lameq_f63,axiom,
    ! [A_27a: del,V1f: $i] :
      ( mem(V1f,arr(A_27a,bool))
     => ! [V0P: $i] :
          ( mem(V0P,arr(A_27a,bool))
         => ! [V3x: $i] : ( ap(f63(A_27a,V1f,V0P),V3x) = ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),f62(A_27a,V1f,V3x)) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__EXISTS__UNIQUE__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,bool))
         => ( p(ap(ap(c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a),V0P),V1f))
          <=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),f61(A_27a,V1f)))
              & p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),f63(A_27a,V1f,V0P))) ) ) ) ) ).

tff(lamtp_f64,type,
    f64: ( del * $i * $i ) > $i ).

tff(lameq_f64,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,bool))
         => ! [V2x: $i] : ( ap(f64(A_27a,V0P,V1f),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P)),ap(V1f,V2x)) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__SELECT__THM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,bool))
         => ( ap(ap(c_2Ebool_2ERES__SELECT(A_27a),V0P),V1f) = ap(c_2Emin_2E_40(A_27a),f64(A_27a,V0P,V1f)) ) ) ) ).

tff(conj_thm_2Ebool_2ERES__FORALL__TRUE,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),k(A_27a,c_2Ebool_2ET)))
      <=> $true ) ) ).

tff(conj_thm_2Ebool_2ERES__EXISTS__FALSE,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),k(A_27a,c_2Ebool_2EF)))
      <=> $false ) ) ).

tff(lamtp_f65,type,
    f65: $i ).

tff(lameq_f65,axiom,
    ! [V4b: tp__o] : ( ap(f65,inj__o(V4b)) = ap(c_2Ebool_2E_7E,inj__o(V4b)) ) ).

tff(conj_thm_2Ebool_2EBOOL__FUN__CASES__THM,axiom,
    ! [V0f: $i] :
      ( mem(V0f,arr(bool,bool))
     => ( ( V0f = k(bool,c_2Ebool_2ET) )
        | ( V0f = k(bool,c_2Ebool_2EF) )
        | ( V0f = i(bool) )
        | ( V0f = f65 ) ) ) ).

tff(conj_thm_2Ebool_2EBOOL__FUN__INDUCT,axiom,
    ! [V0P: $i] :
      ( mem(V0P,arr(arr(bool,bool),bool))
     => ( ( p(ap(V0P,k(bool,c_2Ebool_2ET)))
          & p(ap(V0P,k(bool,c_2Ebool_2EF)))
          & p(ap(V0P,i(bool)))
          & p(ap(V0P,f65)) )
       => ! [V5f: $i] :
            ( mem(V5f,arr(bool,bool))
           => p(ap(V0P,V5f)) ) ) ) ).

tff(conj_thm_2Ebool_2Eliteral__case__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1x: $i] :
          ( mem(V1x,A_27a)
         => ( ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V0f),V1x) = ap(V0f,V1x) ) ) ) ).

tff(lamtp_f66,type,
    f66: ( del * del * del * $i * $i ) > $i ).

tff(lameq_f66,axiom,
    ! [A_27b: del,A_27c: del,A_27a: del,V1N: $i] :
      ( mem(V1N,arr(A_27a,A_27b))
     => ! [V0P: $i] :
          ( mem(V0P,arr(A_27b,A_27c))
         => ! [V4x: $i] : ( ap(f66(A_27b,A_27c,A_27a,V1N,V0P),V4x) = ap(V0P,ap(V1N,V4x)) ) ) ) ).

tff(conj_thm_2Ebool_2Eliteral__case__RAND,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
      ( mem(V0P,arr(A_27b,A_27c))
     => ! [V1N: $i] :
          ( mem(V1N,arr(A_27a,A_27b))
         => ! [V2M: $i] :
              ( mem(V2M,A_27a)
             => ( ap(V0P,ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),f53(A_27b,A_27a,V1N)),V2M)) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27c),f66(A_27b,A_27c,A_27a,V1N,V0P)),V2M) ) ) ) ) ).

tff(conj_thm_2Ebool_2Eliteral__case__RATOR,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0N: $i] :
      ( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
     => ! [V1M: $i] :
          ( mem(V1M,A_27a)
         => ! [V2b: $i] :
              ( mem(V2b,A_27b)
             => ( ap(ap(ap(c_2Ebool_2Eliteral__case(A_27a,arr(A_27b,A_27c)),f55(A_27c,A_27b,A_27a,V0N)),V1M),V2b) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27c),f56(A_27b,A_27c,A_27a,V0N,V2b)),V1M) ) ) ) ) ).

tff(conj_thm_2Ebool_2Eliteral__case__CONG,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,A_27b))
     => ! [V1g: $i] :
          ( mem(V1g,arr(A_27a,A_27b))
         => ! [V2M: $i] :
              ( mem(V2M,A_27a)
             => ! [V3N: $i] :
                  ( mem(V3N,A_27a)
                 => ( ( ( V2M = V3N )
                      & ! [V4x: $i] :
                          ( mem(V4x,A_27a)
                         => ( ( V4x = V3N )
                           => ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) )
                   => ( ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V0f),V2M) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V1g),V3N) ) ) ) ) ) ) ).

tff(lamtp_f67,type,
    f67: ( del * del * $i * $i * $i ) > $i ).

tff(lameq_f67,axiom,
    ! [A_27b: del,A_27a: del,V1t: $i] :
      ( mem(V1t,A_27b)
     => ! [V0a: $i] :
          ( mem(V0a,A_27a)
         => ! [V2u: $i] :
              ( mem(V2u,A_27b)
             => ! [V3x: $i] : ( ap(f67(A_27b,A_27a,V1t,V0a,V2u),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V0a)),V1t),V2u) ) ) ) ) ).

tff(conj_thm_2Ebool_2Eliteral__case__id,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( mem(V0a,A_27a)
     => ! [V1t: $i] :
          ( mem(V1t,A_27b)
         => ! [V2u: $i] :
              ( mem(V2u,A_27b)
             => ( ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),f67(A_27b,A_27a,V1t,V0a,V2u)),V0a) = V1t ) ) ) ) ).

tff(conj_thm_2Ebool_2EBOUNDED__THM,axiom,
    ! [V0v: tp__o] :
      ( p(ap(c_2Ebool_2EBOUNDED,inj__o(V0v)))
    <=> $true ) ).

tff(conj_thm_2Ebool_2ELCOMM__THM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
     => ( ! [V1x: $i] :
            ( mem(V1x,A_27a)
           => ! [V2y: $i] :
                ( mem(V2y,A_27a)
               => ! [V3z: $i] :
                    ( mem(V3z,A_27a)
                   => ( ap(ap(V0f,V1x),ap(ap(V0f,V2y),V3z)) = ap(ap(V0f,ap(ap(V0f,V1x),V2y)),V3z) ) ) ) )
       => ( ! [V4x: $i] :
              ( mem(V4x,A_27a)
             => ! [V5y: $i] :
                  ( mem(V5y,A_27a)
                 => ( ap(ap(V0f,V4x),V5y) = ap(ap(V0f,V5y),V4x) ) ) )
         => ! [V6x: $i] :
              ( mem(V6x,A_27a)
             => ! [V7y: $i] :
                  ( mem(V7y,A_27a)
                 => ! [V8z: $i] :
                      ( mem(V8z,A_27a)
                     => ( ap(ap(V0f,V6x),ap(ap(V0f,V7y),V8z)) = ap(ap(V0f,V7y),ap(ap(V0f,V6x),V8z)) ) ) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EDATATYPE__TAG__THM,axiom,
    ! [A_27a: del,V0x: $i] :
      ( mem(V0x,A_27a)
     => ( p(ap(c_2Ebool_2EDATATYPE(A_27a),V0x))
      <=> $true ) ) ).

tff(conj_thm_2Ebool_2EDATATYPE__BOOL,axiom,
    ! [V0bool: $i] :
      ( mem(V0bool,arr(bool,arr(bool,bool)))
     => ( p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(V0bool,inj__o(fo__c_2Ebool_2ET)),inj__o(fo__c_2Ebool_2EF))))
      <=> $true ) ) ).

tff(ax_thm_2Ebool_2Eitself__TY__DEF,axiom,
    ! [A_27a: del] :
    ? [V0rep: $i] :
      ( mem(V0rep,arr(ty_2Ebool_2Eitself(A_27a),A_27a))
      & p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,ty_2Ebool_2Eitself(A_27a)),ap(c_2Emin_2E_3D(A_27a),c_2Ebool_2EARB(A_27a))),V0rep)) ) ).

tff(conj_thm_2Ebool_2EITSELF__UNIQUE,axiom,
    ! [A_27a: del,V0i: $i] :
      ( mem(V0i,ty_2Ebool_2Eitself(A_27a))
     => ( V0i = c_2Ebool_2Ethe__value(A_27a) ) ) ).

tff(conj_thm_2Ebool_2Eitself__Axiom,axiom,
    ! [A_27a: del,A_27b: del,V0e: $i] :
      ( mem(V0e,A_27b)
     => ? [V1f: $i] :
          ( mem(V1f,arr(ty_2Ebool_2Eitself(A_27a),A_27b))
          & ( ap(V1f,c_2Ebool_2Ethe__value(A_27a)) = V0e ) ) ) ).

tff(conj_thm_2Ebool_2Eitself__induction,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
     => ( p(ap(V0P,c_2Ebool_2Ethe__value(A_27a)))
       => ! [V1i: $i] :
            ( mem(V1i,ty_2Ebool_2Eitself(A_27a))
           => p(ap(V0P,V1i)) ) ) ) ).

tff(ax_thm_2Ebool_2Eitself__case__thm,axiom,
    ! [A_27a: del,A_27b: del,V0b: $i] :
      ( mem(V0b,A_27b)
     => ( ap(ap(c_2Ebool_2Eitself__case(A_27a,A_27b),c_2Ebool_2Ethe__value(A_27a)),V0b) = V0b ) ) ).

tff(conj_thm_2Ebool_2EFORALL__itself,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
     => ( ! [V1x: $i] :
            ( mem(V1x,ty_2Ebool_2Eitself(A_27a))
           => p(ap(V0P,V1x)) )
      <=> p(ap(V0P,c_2Ebool_2Ethe__value(A_27a))) ) ) ).

tff(conj_thm_2Ebool_2EEXISTS__itself,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
     => ( ? [V1x: $i] :
            ( mem(V1x,ty_2Ebool_2Eitself(A_27a))
            & p(ap(V0P,V1x)) )
      <=> p(ap(V0P,c_2Ebool_2Ethe__value(A_27a))) ) ) ).

tff(conj_thm_2Ebool_2EPULL__EXISTS,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ( ( ? [V2x: $i] :
                  ( mem(V2x,A_27a)
                  & p(ap(V0P,V2x)) )
             => p(inj__o(V1Q)) )
          <=> ! [V3x: $i] :
                ( mem(V3x,A_27a)
               => ( p(ap(V0P,V3x))
                 => p(inj__o(V1Q)) ) ) )
          & ( ( ? [V4x: $i] :
                  ( mem(V4x,A_27a)
                  & p(ap(V0P,V4x)) )
              & p(inj__o(V1Q)) )
          <=> ? [V5x: $i] :
                ( mem(V5x,A_27a)
                & p(ap(V0P,V5x))
                & p(inj__o(V1Q)) ) )
          & ( ( p(inj__o(V1Q))
              & ? [V6x: $i] :
                  ( mem(V6x,A_27a)
                  & p(ap(V0P,V6x)) ) )
          <=> ? [V7x: $i] :
                ( mem(V7x,A_27a)
                & p(inj__o(V1Q))
                & p(ap(V0P,V7x)) ) ) ) ) ).

tff(conj_thm_2Ebool_2EPULL__FORALL,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1Q: tp__o] :
          ( ( ( p(inj__o(V1Q))
             => ! [V2x: $i] :
                  ( mem(V2x,A_27a)
                 => p(ap(V0P,V2x)) ) )
          <=> ! [V3x: $i] :
                ( mem(V3x,A_27a)
               => ( p(inj__o(V1Q))
                 => p(ap(V0P,V3x)) ) ) )
          & ( ( ! [V4x: $i] :
                  ( mem(V4x,A_27a)
                 => p(ap(V0P,V4x)) )
              & p(inj__o(V1Q)) )
          <=> ! [V5x: $i] :
                ( mem(V5x,A_27a)
               => ( p(ap(V0P,V5x))
                  & p(inj__o(V1Q)) ) ) )
          & ( ( p(inj__o(V1Q))
              & ! [V6x: $i] :
                  ( mem(V6x,A_27a)
                 => p(ap(V0P,V6x)) ) )
          <=> ! [V7x: $i] :
                ( mem(V7x,A_27a)
               => ( p(inj__o(V1Q))
                  & p(ap(V0P,V7x)) ) ) ) ) ) ).

tff(conj_thm_2Ebool_2EPEIRCE,axiom,
    ! [V0P: tp__o,V1Q: tp__o] :
      ( ( ( p(inj__o(V0P))
         => p(inj__o(V1Q)) )
       => p(inj__o(V0P)) )
     => p(inj__o(V0P)) ) ).

tff(conj_thm_2Ebool_2EJRH__INDUCT__UTIL,axiom,
    ! [A_27a: del,V0P: $i] :
      ( mem(V0P,arr(A_27a,bool))
     => ! [V1t: $i] :
          ( mem(V1t,A_27a)
         => ( ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( ( V2x = V1t )
                 => p(ap(V0P,V2x)) ) )
           => p(ap(c_2Ebool_2E_3F(A_27a),V0P)) ) ) ) ).

tff(conj_thm_2Ebool_2EDISJ__EQ__IMP,axiom,
    ! [V0A: tp__o,V1B: tp__o] :
      ( ( p(inj__o(V0A))
        | p(inj__o(V1B)) )
    <=> ( ~ p(inj__o(V0A))
       => p(inj__o(V1B)) ) ) ).

%------------------------------------------------------------------------------