ITP001 Axioms: ITP003^5.ax
%------------------------------------------------------------------------------
% File : ITP003^5 : TPTP v8.2.0. 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 : 248 ( 25 unt; 29 typ; 0 def)
% Number of atoms : 2399 ( 106 equ; 0 cnn)
% Maximal formula atoms : 31 ( 9 avg)
% Number of connectives : 4414 ( 46 ~; 61 |; 182 &;3405 @)
% ( 151 <=>; 569 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 10 avg;3405 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 39 usr; 19 con; 0-2 aty)
% Number of variables : 747 ( 90 ^ 601 !; 56 ?; 747 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Ebool_2Eitself,type,
ty_2Ebool_2Eitself: del > del ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(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 ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(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 ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
thf(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(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 ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21: del > $i ).
thf(mem_c_2Ebool_2E_3F_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(tp_c_2Ebool_2EARB,type,
c_2Ebool_2EARB: del > $i ).
thf(mem_c_2Ebool_2EARB,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2EARB @ A_27a ) @ A_27a ) ).
thf(tp_c_2Ebool_2EBOUNDED,type,
c_2Ebool_2EBOUNDED: $i ).
thf(mem_c_2Ebool_2EBOUNDED,axiom,
mem @ c_2Ebool_2EBOUNDED @ ( arr @ bool @ bool ) ).
thf(tp_c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND: del > $i ).
thf(mem_c_2Ebool_2ECOND,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2ECOND @ A_27a ) @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE: del > $i ).
thf(mem_c_2Ebool_2EDATATYPE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2EDATATYPE @ A_27a ) @ ( arr @ A_27a @ bool ) ) ).
thf(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
thf(mem_c_2Ebool_2EF,axiom,
mem @ c_2Ebool_2EF @ bool ).
thf(ax_false_p,axiom,
~ ( p @ c_2Ebool_2EF ) ).
thf(tp_c_2Ebool_2EIN,type,
c_2Ebool_2EIN: del > $i ).
thf(mem_c_2Ebool_2EIN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2EIN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ebool_2ELET,type,
c_2Ebool_2ELET: del > del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE: del > del > $i ).
thf(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 ) ) ).
thf(tp_c_2Ebool_2EONTO,type,
c_2Ebool_2EONTO: del > del > $i ).
thf(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 ) ) ).
thf(tp_c_2Ebool_2ERES__ABSTRACT,type,
c_2Ebool_2ERES__ABSTRACT: del > del > $i ).
thf(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 ) ) ) ) ).
thf(tp_c_2Ebool_2ERES__EXISTS,type,
c_2Ebool_2ERES__EXISTS: del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2ERES__EXISTS__UNIQUE,type,
c_2Ebool_2ERES__EXISTS__UNIQUE: del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2ERES__FORALL,type,
c_2Ebool_2ERES__FORALL: del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2ERES__SELECT,type,
c_2Ebool_2ERES__SELECT: del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
thf(mem_c_2Ebool_2ET,axiom,
mem @ c_2Ebool_2ET @ bool ).
thf(ax_true_p,axiom,
p @ c_2Ebool_2ET ).
thf(tp_c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION: del > del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
thf(mem_c_2Ebool_2E_5C_2F,axiom,
mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(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 ) ) ) ) ) ).
thf(tp_c_2Ebool_2Eitself__case,type,
c_2Ebool_2Eitself__case: del > del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2Eliteral__case,type,
c_2Ebool_2Eliteral__case: del > del > $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ebool_2Ethe__value,type,
c_2Ebool_2Ethe__value: del > $i ).
thf(mem_c_2Ebool_2Ethe__value,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2Ethe__value @ A_27a ) @ ( ty_2Ebool_2Eitself @ A_27a ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(ax_thm_2Ebool_2ET__DEF,axiom,
( $true
<=> ( ( lam @ bool
@ ^ [V0x: $i] : V0x )
= ( lam @ bool
@ ^ [V1x: $i] : V1x ) ) ) ).
thf(ax_thm_2Ebool_2EFORALL__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2E_21 @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0P: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V0P )
@ ( lam @ A_27a
@ ^ [V1x: $i] : c_2Ebool_2ET ) ) ) ) ).
thf(ax_thm_2Ebool_2EEXISTS__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2E_3F @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0P: $i] : ( ap @ V0P @ ( ap @ ( c_2Emin_2E_40 @ A_27a ) @ V0P ) ) ) ) ).
thf(ax_thm_2Ebool_2EAND__DEF,axiom,
( c_2Ebool_2E_2F_5C
= ( lam @ bool
@ ^ [V0t1: $i] :
( lam @ bool
@ ^ [V1t2: $i] :
( ap @ ( c_2Ebool_2E_21 @ bool )
@ ( lam @ bool
@ ^ [V2t: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ V0t1 ) @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ V1t2 ) @ V2t ) ) ) @ V2t ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EOR__DEF,axiom,
( c_2Ebool_2E_5C_2F
= ( lam @ bool
@ ^ [V0t1: $i] :
( lam @ bool
@ ^ [V1t2: $i] :
( ap @ ( c_2Ebool_2E_21 @ bool )
@ ( lam @ bool
@ ^ [V2t: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ V0t1 ) @ V2t ) ) @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ V1t2 ) @ V2t ) ) @ V2t ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EF__DEF,axiom,
( $false
<=> ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( p @ V0t ) ) ) ).
thf(ax_thm_2Ebool_2ENOT__DEF,axiom,
( c_2Ebool_2E_7E
= ( lam @ bool
@ ^ [V0t: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ V0t ) @ c_2Ebool_2EF ) ) ) ).
thf(ax_thm_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2E_3F_21 @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0P: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V0P ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2ELET__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Ebool_2ELET @ A_27a @ A_27b )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V0f: $i] :
( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0f @ V1x ) ) ) ) ).
thf(ax_thm_2Ebool_2ECOND__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2ECOND @ A_27a )
= ( lam @ bool
@ ^ [V0t: $i] :
( lam @ A_27a
@ ^ [V1t1: $i] :
( lam @ A_27a
@ ^ [V2t2: $i] :
( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ V0t ) @ 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 ) @ V0t ) @ c_2Ebool_2EF ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V2t2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EONE__ONE__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Ebool_2EONE__ONE @ A_27a @ A_27b )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V0f: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x1: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x2: $i] : ( 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 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EONTO__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Ebool_2EONTO @ A_27a @ A_27b )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V0f: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V1y: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V1y ) @ ( ap @ V0f @ V2x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2ETYPE__DEFINITION,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0P: $i] :
( lam @ ( arr @ A_27b @ A_27a )
@ ^ [V1rep: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V2x_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V3x_27_27: $i] : ( 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 ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V0P @ V4x ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V5x_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4x ) @ ( ap @ V1rep @ V5x_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
<=> $true )
| ( ( p @ V0t )
<=> $false ) ) ) ).
thf(ax_thm_2Ebool_2EETA__AX,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ A_27b ) )
=> ( ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0t @ V1x ) )
= V0t ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(stp_i,type,
tp__i: $tType ).
thf(stp_inj_i,type,
inj__i: tp__i > $i ).
thf(stp_surj_i,type,
surj__i: $i > tp__i ).
thf(stp_inj_surj_i,axiom,
! [X: tp__i] :
( ( surj__i @ ( inj__i @ X ) )
= X ) ).
thf(stp_inj_mem_i,axiom,
! [X: tp__i] : ( mem @ ( inj__i @ X ) @ ind ) ).
thf(stp_iso_mem_i,axiom,
! [X: $i] :
( ( mem @ X @ ind )
=> ( X
= ( inj__i @ ( surj__i @ X ) ) ) ) ).
thf(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 ) ) ) ).
thf(ax_thm_2Ebool_2Eliteral__case__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V0f: $i] :
( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0f @ V1x ) ) ) ) ).
thf(ax_thm_2Ebool_2EIN__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2EIN @ A_27a )
= ( lam @ A_27a
@ ^ [V0x: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1f: $i] : ( ap @ V1f @ V0x ) ) ) ) ).
thf(ax_thm_2Ebool_2ERES__FORALL__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2ERES__FORALL @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0p: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1m: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0p ) ) @ ( ap @ V1m @ V2x ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2ERES__EXISTS__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0p: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1m: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0p ) ) @ ( ap @ V1m @ V2x ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2ERES__EXISTS__UNIQUE__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0p: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1m: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1m @ V2x ) ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2ERES__SELECT__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2ERES__SELECT @ A_27a )
= ( lam @ ( arr @ A_27a @ bool )
@ ^ [V0p: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1m: $i] :
( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0p ) ) @ ( ap @ V1m @ V2x ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EBOUNDED__DEF,axiom,
( c_2Ebool_2EBOUNDED
= ( lam @ bool
@ ^ [V0v: $i] : c_2Ebool_2ET ) ) ).
thf(ax_thm_2Ebool_2EDATATYPE__TAG__DEF,axiom,
! [A_27a: del] :
( ( c_2Ebool_2EDATATYPE @ A_27a )
= ( lam @ A_27a
@ ^ [V0x: $i] : c_2Ebool_2ET ) ) ).
thf(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
thf(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
=> ( p @ V1t2 ) )
=> ( ( ( p @ V1t2 )
=> ( p @ V0t1 ) )
=> ( ( p @ V0t1 )
<=> ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( $false
=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EETA__THM,axiom,
! [A_27a: del,A_27b: del,V0M: $i] :
( ( mem @ V0M @ ( arr @ A_27a @ A_27b ) )
=> ( ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0M @ V1x ) )
= V0M ) ) ).
thf(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( p @ V0t )
| ~ ( p @ V0t ) ) ) ).
thf(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
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0f @ V2x ) )
@ V1y )
= ( ap @ V0f @ V1y ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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
@ ( lam @ A_27b
@ ^ [V2x: $i] : V0t1 )
@ V1t2 )
= V0t1 ) ) ) ).
thf(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EAND__INTRO__THM,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( p @ V0t1 )
=> ( ( p @ V1t2 )
=> ( ( p @ V0t1 )
& ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND1__THM,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V0t1 ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND2__THM,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V1t2 ) ) ) ) ).
thf(conj_thm_2Ebool_2ECONJ__SYM,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
<=> ( ( p @ V1t2 )
& ( p @ V0t1 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECONJ__COMM,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
<=> ( ( p @ V1t2 )
& ( p @ V0t1 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECONJ__ASSOC,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ bool )
=> ( ( ( p @ V0t1 )
& ( p @ V1t2 )
& ( p @ V2t3 ) )
<=> ( ( p @ V0t1 )
& ( p @ V1t2 )
& ( p @ V2t3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__INTRO__THM1,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( p @ V0t1 )
=> ( ( p @ V0t1 )
| ( p @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__INTRO__THM2,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( p @ V1t2 )
=> ( ( p @ V0t1 )
| ( p @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__ELIM__THM,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ bool )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ bool )
=> ( ( ( p @ V1t1 )
| ( p @ V2t2 ) )
=> ( ( ( p @ V1t1 )
=> ( p @ V0t ) )
=> ( ( ( p @ V2t2 )
=> ( p @ V0t ) )
=> ( p @ V0t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $false )
=> ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ( p @ V0t )
=> ( ( p @ V0t )
=> $false ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__F,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ( p @ V0t )
=> ( ( p @ V0t )
<=> $false ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__AND,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ~ ( ( p @ V0t )
& ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
& ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
& $true )
<=> ( p @ V0t ) )
& ( ( $false
& ( p @ V0t ) )
<=> $false )
& ( ( ( p @ V0t )
& $false )
<=> $false )
& ( ( ( p @ V0t )
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
| ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
| $true )
<=> $true )
& ( ( $false
| ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
| $false )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
| ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
=> $true )
<=> $true )
& ( ( $false
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__REFL,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( V0x = V0x ) ) ).
thf(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( V0x = V0x )
<=> $true ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOOL__EQ__DISTINCT,axiom,
( ~ ( $true
<=> $false )
& ~ ( $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
<=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $true )
<=> ( p @ V0t ) )
& ( ( $false
<=> ( p @ V0t ) )
<=> ~ ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(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 ) @ c_2Ebool_2ET ) @ V0t1 ) @ V1t2 )
= V0t1 )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ c_2Ebool_2EF ) @ V0t1 ) @ V1t2 )
= V1t2 ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__ID,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t: $i] :
( ( mem @ V1t @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V0b ) @ V1t ) @ V1t )
= V1t ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0P @ V1x ) ) ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ESELECT__REFL,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1y ) @ V0x ) ) )
= V0x ) ) ).
thf(conj_thm_2Ebool_2ESELECT__REFL__2,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) @ V1y ) ) )
= V0x ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ V2x ) ) )
& ( p @ V1Q ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
& ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ V0P )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ V0P )
& ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) )
| ( p @ V1Q ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( ( p @ ( ap @ V0P @ V3x ) )
| ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ V0P )
| ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( ( p @ V0P )
| ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOTH__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ V0P )
& ( p @ V1Q ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ V0P ) )
& ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) )
& ( p @ V1Q ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V0P @ V3x ) ) )
& ( p @ V1Q ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ V0P )
& ( p @ ( ap @ V1Q @ V2x ) ) )
<=> ( ( p @ V0P )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOTH__FORALL__OR__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ V0P )
| ( p @ V1Q ) ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ V0P ) )
| ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
! [A_27a: del,V0Q: $i] :
( ( mem @ V0Q @ bool )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V1P @ V2x ) )
| ( p @ V0Q ) ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ V1P @ V3x ) ) )
| ( p @ V0Q ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ V0P )
| ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ( ( p @ V0P )
| ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOTH__FORALL__IMP__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ V0P )
=> ( p @ V1Q ) ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ V0P ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__FORALL__IMP__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2x ) )
=> ( p @ V1Q ) ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V0P @ V3x ) ) )
=> ( p @ V1Q ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__FORALL__IMP__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ V0P )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ( ( p @ V0P )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOTH__EXISTS__IMP__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( ( p @ V0P )
=> ( p @ V1Q ) ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ V0P ) )
=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__EXISTS__IMP__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( ( p @ ( ap @ V0P @ V2x ) )
=> ( p @ V1Q ) ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ V0P @ V3x ) ) )
=> ( p @ V1Q ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__EXISTS__IMP__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( ( p @ V0P )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ( ( p @ V0P )
=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__IMP__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) )
<=> ( ( p @ V1B )
=> ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ~ ( ( p @ V0A )
=> ( p @ V1B ) )
<=> ( ( p @ V0A )
& ~ ( p @ V1B ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) )
<=> ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__COMM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
& ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
| ~ ( p @ V1B ) ) )
& ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
& ~ ( p @ V1B ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__AND__OVER__OR,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( p @ V0A )
& ( ( p @ V1B )
| ( p @ V2C ) ) )
<=> ( ( ( p @ V0A )
& ( p @ V1B ) )
| ( ( p @ V0A )
& ( p @ V2C ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__AND__OVER__OR,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( ( p @ V1B )
| ( p @ V2C ) )
& ( p @ V0A ) )
<=> ( ( ( p @ V1B )
& ( p @ V0A ) )
| ( ( p @ V2C )
& ( p @ V0A ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__OR__OVER__AND,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( p @ V0A )
| ( ( p @ V1B )
& ( p @ V2C ) ) )
<=> ( ( ( p @ V0A )
| ( p @ V1B ) )
& ( ( p @ V0A )
| ( p @ V2C ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( ( p @ V1B )
& ( p @ V2C ) )
| ( p @ V0A ) )
<=> ( ( ( p @ V1B )
| ( p @ V0A ) )
& ( ( p @ V2C )
| ( p @ V0A ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
=> ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
| ( p @ V1B ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__IMP__THM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ! [V2R: $i] :
( ( mem @ V2R @ bool )
=> ( ( ( ( p @ V0P )
| ( p @ V1Q ) )
=> ( p @ V2R ) )
<=> ( ( ( p @ V0P )
=> ( p @ V2R ) )
& ( ( p @ V1Q )
=> ( p @ V2R ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CONJ__THM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ! [V2R: $i] :
( ( mem @ V2R @ bool )
=> ( ( ( p @ V0P )
=> ( ( p @ V1Q )
& ( p @ V2R ) ) )
<=> ( ( ( p @ V0P )
=> ( p @ V1Q ) )
& ( ( p @ V0P )
=> ( p @ V2R ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__F__EQ__F,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $false )
<=> ( ( p @ V0t )
<=> $false ) ) ) ).
thf(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ bool )
=> ( ( ( p @ V0t1 )
=> ( ( p @ V1t2 )
=> ( p @ V2t3 ) ) )
<=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V2t3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__IMP__THM,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
<=> ( p @ V1t2 ) )
<=> ( ( ( p @ V0t1 )
=> ( p @ V1t2 ) )
& ( ( p @ V1t2 )
=> ( p @ V0t1 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__EXPAND,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
<=> ( p @ V1t2 ) )
<=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
| ( ~ ( p @ V0t1 )
& ~ ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__RATOR,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [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 ) ) @ V0b ) @ V1f ) @ V2g ) @ V3x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V0b ) @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__RAND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ bool )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ap @ V0f @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V1b ) @ V2x ) @ V3y ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V1b ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ V0f @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__ABS,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27b ) )
=> ( ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V0b ) @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ A_27b ) ) @ V0b ) @ V1f ) @ V2g ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__EXPAND,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ bool )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0b ) @ V1t1 ) @ V2t2 ) )
<=> ( ( ~ ( p @ V0b )
| ( p @ V1t1 ) )
& ( ( p @ V0b )
| ( p @ V2t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__EXPAND__IMP,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ bool )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0b ) @ V1t1 ) @ V2t2 ) )
<=> ( ( ( p @ V0b )
=> ( p @ V1t1 ) )
& ( ~ ( p @ V0b )
=> ( p @ V2t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__EXPAND__OR,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ bool )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0b ) @ V1t1 ) @ V2t2 ) )
<=> ( ( ( p @ V0b )
& ( p @ V1t1 ) )
| ( ~ ( p @ V0b )
& ( p @ V2t2 ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V1N @ V3x ) ) )
@ V2M ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ A_27a @ bool )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0P @ ( ap @ V1N @ V4x ) ) ) )
@ V2M ) ) ) ) ) ) ).
thf(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 ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V0N @ V3x ) ) )
@ V1M )
@ V2b )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ A_27a @ A_27c )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ V0N @ V4x ) @ V2b ) ) )
@ V1M ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0P @ V1x ) ) ) )
<=> ( ? [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 ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1x_27: $i] :
( ( mem @ V1x_27 @ bool )
=> ! [V2y: $i] :
( ( mem @ V2y @ bool )
=> ! [V3y_27: $i] :
( ( mem @ V3y_27 @ bool )
=> ( ( ( ( p @ V0x )
<=> ( p @ V1x_27 ) )
& ( ( p @ V1x_27 )
=> ( ( p @ V2y )
<=> ( p @ V3y_27 ) ) ) )
=> ( ( ( p @ V0x )
=> ( p @ V2y ) )
<=> ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND__CONG,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V2Q )
=> ( ( p @ V0P )
<=> ( p @ V1P_27 ) ) )
& ( ( p @ V1P_27 )
=> ( ( p @ V2Q )
<=> ( p @ V3Q_27 ) ) ) )
=> ( ( ( p @ V0P )
& ( p @ V2Q ) )
<=> ( ( p @ V1P_27 )
& ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__AND__CONG,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V0P )
<=> ( p @ V1P_27 ) )
& ( ( p @ V1P_27 )
=> ( ( p @ V2Q )
<=> ( p @ V3Q_27 ) ) ) )
=> ( ( ( p @ V0P )
& ( p @ V2Q ) )
<=> ( ( p @ V1P_27 )
& ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__CONG,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ~ ( p @ V2Q )
=> ( ( p @ V0P )
<=> ( p @ V1P_27 ) ) )
& ( ~ ( p @ V1P_27 )
=> ( ( p @ V2Q )
<=> ( p @ V3Q_27 ) ) ) )
=> ( ( ( p @ V0P )
| ( p @ V2Q ) )
<=> ( ( p @ V1P_27 )
| ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__OR__CONG,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V0P )
<=> ( p @ V1P_27 ) )
& ( ~ ( p @ V1P_27 )
=> ( ( p @ V2Q )
<=> ( p @ V3Q_27 ) ) ) )
=> ( ( ( p @ V0P )
| ( p @ V2Q ) )
<=> ( ( p @ V1P_27 )
| ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__CONG,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ! [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 @ V0P )
<=> ( p @ V1Q ) )
& ( ( p @ V1Q )
=> ( V2x = V3x_27 ) )
& ( ~ ( p @ V1Q )
=> ( V4y = V5y_27 ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V0P ) @ V2x ) @ V4y )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V1Q ) @ V3x_27 ) @ V5y_27 ) ) ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EMONO__AND,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1y: $i] :
( ( mem @ V1y @ bool )
=> ! [V2z: $i] :
( ( mem @ V2z @ bool )
=> ! [V3w: $i] :
( ( mem @ V3w @ bool )
=> ( ( ( ( p @ V0x )
=> ( p @ V1y ) )
& ( ( p @ V2z )
=> ( p @ V3w ) ) )
=> ( ( ( p @ V0x )
& ( p @ V2z ) )
=> ( ( p @ V1y )
& ( p @ V3w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EMONO__OR,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1y: $i] :
( ( mem @ V1y @ bool )
=> ! [V2z: $i] :
( ( mem @ V2z @ bool )
=> ! [V3w: $i] :
( ( mem @ V3w @ bool )
=> ( ( ( ( p @ V0x )
=> ( p @ V1y ) )
& ( ( p @ V2z )
=> ( p @ V3w ) ) )
=> ( ( ( p @ V0x )
| ( p @ V2z ) )
=> ( ( p @ V1y )
| ( p @ V3w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EMONO__IMP,axiom,
! [V0y: $i] :
( ( mem @ V0y @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ! [V2z: $i] :
( ( mem @ V2z @ bool )
=> ! [V3w: $i] :
( ( mem @ V3w @ bool )
=> ( ( ( ( p @ V0y )
=> ( p @ V1x ) )
& ( ( p @ V2z )
=> ( p @ V3w ) ) )
=> ( ( ( p @ V1x )
=> ( p @ V2z ) )
=> ( ( p @ V0y )
=> ( p @ V3w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EMONO__NOT,axiom,
! [V0y: $i] :
( ( mem @ V0y @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( ( p @ V0y )
=> ( p @ V1x ) )
=> ( ~ ( p @ V1x )
=> ~ ( p @ V0y ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EMONO__NOT__EQ,axiom,
! [V0y: $i] :
( ( mem @ V0y @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( ( p @ V0y )
=> ( p @ V1x ) )
<=> ( ~ ( p @ V1x )
=> ~ ( p @ V0y ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EMONO__COND,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1y: $i] :
( ( mem @ V1y @ bool )
=> ! [V2z: $i] :
( ( mem @ V2z @ bool )
=> ! [V3w: $i] :
( ( mem @ V3w @ bool )
=> ! [V4b: $i] :
( ( mem @ V4b @ bool )
=> ( ( ( p @ V0x )
=> ( p @ V1y ) )
=> ( ( ( p @ V2z )
=> ( p @ V3w ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V4b ) @ V0x ) @ V2z ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V4b ) @ V1y ) @ V3w ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EEXISTS__REFL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V1x = V0a ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ V0a ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 ) @ 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 ) @ c_2Ebool_2EF ) @ V2t1 ) @ V3t2 )
= V3t2 ) ) ) ) ).
thf(conj_thm_2Ebool_2Ebool__case__ID,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t: $i] :
( ( mem @ V1t @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V0b ) @ V1t ) @ V1t )
= V1t ) ) ) ).
thf(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 @ c_2Ebool_2ET )
= V0t1 )
& ( ( ap @ V2fn @ c_2Ebool_2EF )
= V1t2 ) ) ) ) ).
thf(conj_thm_2Ebool_2Ebool__INDUCT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ bool @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ c_2Ebool_2ET ) )
& ( p @ ( ap @ V0P @ c_2Ebool_2EF ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ bool )
=> ( p @ ( ap @ V0P @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebool_2Ebool__case__CONG,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ! [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 @ V0P )
<=> ( p @ V1Q ) )
& ( ( p @ V1Q )
=> ( V2x = V3x_27 ) )
& ( ~ ( p @ V1Q )
=> ( V4y = V5y_27 ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V0P ) @ V2x ) @ V4y )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V1Q ) @ V3x_27 ) @ V5y_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EFORALL__BOOL,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ bool @ bool ) )
=> ( ! [V1b: $i] :
( ( mem @ V1b @ bool )
=> ( p @ ( ap @ V0P @ V1b ) ) )
<=> ( ( p @ ( ap @ V0P @ c_2Ebool_2ET ) )
& ( p @ ( ap @ V0P @ c_2Ebool_2EF ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V0P @ V2x ) ) @ ( ap @ V1Q @ V2x ) ) ) ) )
=> ( ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V0P @ V3x ) ) ) )
| ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V1Q @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EUEXISTS__SIMP,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V0t ) ) )
<=> ( ( p @ V0t )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( V2x = V3y ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1f @ V2x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) ) @ ( ap @ V1f @ V2x ) ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] : c_2Ebool_2ET ) ) )
<=> $true ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] : c_2Ebool_2EF ) ) )
<=> $false ) ) ).
thf(conj_thm_2Ebool_2EBOOL__FUN__CASES__THM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ bool @ bool ) )
=> ( ( V0f
= ( lam @ bool
@ ^ [V1b: $i] : c_2Ebool_2ET ) )
| ( V0f
= ( lam @ bool
@ ^ [V2b: $i] : c_2Ebool_2EF ) )
| ( V0f
= ( lam @ bool
@ ^ [V3b: $i] : V3b ) )
| ( V0f
= ( lam @ bool
@ ^ [V4b: $i] : ( ap @ c_2Ebool_2E_7E @ V4b ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOOL__FUN__INDUCT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ bool @ bool ) @ bool ) )
=> ( ( ( p
@ ( ap @ V0P
@ ( lam @ bool
@ ^ [V1b: $i] : c_2Ebool_2ET ) ) )
& ( p
@ ( ap @ V0P
@ ( lam @ bool
@ ^ [V2b: $i] : c_2Ebool_2EF ) ) )
& ( p
@ ( ap @ V0P
@ ( lam @ bool
@ ^ [V3b: $i] : V3b ) ) )
& ( p
@ ( ap @ V0P
@ ( lam @ bool
@ ^ [V4b: $i] : ( ap @ c_2Ebool_2E_7E @ V4b ) ) ) ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ bool @ bool ) )
=> ( p @ ( ap @ V0P @ V5f ) ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V1N @ V3x ) ) )
@ V2M ) )
= ( ap
@ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27c )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0P @ ( ap @ V1N @ V4x ) ) ) )
@ V2M ) ) ) ) ) ).
thf(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 ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V0N @ V3x ) ) )
@ V1M )
@ V2b )
= ( ap
@ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27c )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ V0N @ V4x ) @ V2b ) ) )
@ V1M ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V0a ) ) @ V1t ) @ V2u ) ) )
@ V0a )
= V1t ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOUNDED__THM,axiom,
! [V0v: $i] :
( ( mem @ V0v @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2EBOUNDED @ V0v ) )
<=> $true ) ) ).
thf(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 ) ) ) ) ) ) ) ) ) ).
thf(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 ) ) ).
thf(conj_thm_2Ebool_2EDATATYPE__BOOL,axiom,
! [V0bool: $i] :
( ( mem @ V0bool @ ( arr @ bool @ ( arr @ bool @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ V0bool @ c_2Ebool_2ET ) @ c_2Ebool_2EF ) ) )
<=> $true ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EPULL__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) )
=> ( p @ V1Q ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ V1Q ) ) ) )
& ( ( ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ V0P @ V4x ) ) )
& ( p @ V1Q ) )
<=> ? [V5x: $i] :
( ( mem @ V5x @ A_27a )
& ( p @ ( ap @ V0P @ V5x ) )
& ( p @ V1Q ) ) )
& ( ( ( p @ V1Q )
& ? [V6x: $i] :
( ( mem @ V6x @ A_27a )
& ( p @ ( ap @ V0P @ V6x ) ) ) )
<=> ? [V7x: $i] :
( ( mem @ V7x @ A_27a )
& ( p @ V1Q )
& ( p @ ( ap @ V0P @ V7x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EPULL__FORALL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ( ( p @ V1Q )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ V2x ) ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ V1Q )
=> ( p @ ( ap @ V0P @ V3x ) ) ) ) )
& ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ V0P @ V4x ) ) )
& ( p @ V1Q ) )
<=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V5x ) )
& ( p @ V1Q ) ) ) )
& ( ( ( p @ V1Q )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( p @ ( ap @ V0P @ V6x ) ) ) )
<=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ V1Q )
& ( p @ ( ap @ V0P @ V7x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EPEIRCE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ( ( p @ V0P )
=> ( p @ V1Q ) )
=> ( p @ V0P ) )
=> ( p @ V0P ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__EQ__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
=> ( p @ V1B ) ) ) ) ) ).
%------------------------------------------------------------------------------