ITP001 Axioms: ITP032^5.ax
%------------------------------------------------------------------------------
% File : ITP032^5 : 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 : res_quan^2.ax [Gau20]
% : HL4032^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 0 unt; 0 typ; 0 def)
% Number of atoms : 1125 ( 19 equ; 0 cnn)
% Maximal formula atoms : 36 ( 19 avg)
% Number of connectives : 1886 ( 12 ~; 4 |; 13 &;1670 @)
% ( 38 <=>; 149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 16 avg;1670 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 33 ( 32 usr; 32 con; 0-2 aty)
% Number of variables : 287 ( 67 ^ 216 !; 4 ?; 287 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(conj_thm_2Eres__quan_2ERES__FORALL,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_2Eres__quan_2ERES__EXISTS,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_2Eres__quan_2ERES__EXISTS__UNIQUE,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_2Eres__quan_2ERES__SELECT,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_2Eres__quan_2ERES__FORALL__CONJ__DIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1Q @ V3i ) ) @ ( ap @ V2R @ V3i ) ) ) ) )
<=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ V1Q @ V4i ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ V2R @ V5i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__DISJ__DIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a )
@ ( lam @ A_27a
@ ^ [V3j: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V0P @ V3j ) ) @ ( ap @ V1Q @ V3j ) ) ) )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ V2R @ V4i ) ) ) )
<=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ V2R @ V5i ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1Q )
@ ( lam @ A_27a
@ ^ [V6i: $i] : ( ap @ V2R @ V6i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__UNIQUE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1j: $i] :
( ( mem @ V1j @ A_27a )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1j ) )
@ ( lam @ A_27a
@ ^ [V2i: $i] : ( ap @ V0P @ V2i ) ) ) )
<=> ( p @ ( ap @ V0P @ V1j ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__FORALL,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ ( ap @ V1R @ V4i ) @ V3x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V5i: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V6x: $i] : ( ap @ ( ap @ V1R @ V5i ) @ V6x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__REORDER,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V3i: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27b ) @ V1Q )
@ ( lam @ A_27b
@ ^ [V4j: $i] : ( ap @ ( ap @ V2R @ V3i ) @ V4j ) ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27b ) @ V1Q )
@ ( lam @ A_27b
@ ^ [V5j: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V6i: $i] : ( ap @ ( ap @ V2R @ V6i ) @ V5j ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__T,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27b ) @ V1s )
@ ( lam @ A_27b
@ ^ [V3x: $i] : c_2Ebool_2ET ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__F,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27b ) @ V1s )
@ ( lam @ A_27b
@ ^ [V3x: $i] : c_2Ebool_2EF ) ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0p ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__UNIV,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0p ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ V0p ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__NULL,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ bool )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V2x: $i] : V1m ) ) )
<=> ( ( V0p
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( p @ V1m ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ENOT__RES__FORALL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__NOT__EMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1s ) @ V0P ) )
=> ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V2t ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1s ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0P ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1s ) @ V0P ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V2t ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__DIFF,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0P @ V4x ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2t ) ) ) @ ( ap @ V0P @ V5x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2EIN__BIGINTER__RES__FORALL,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1sos ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ V1sos )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__BIGUNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1sos ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ V1sos )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3s: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V3s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0P @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__FORALL__BIGINTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1sos ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ V1sos )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V4s: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V4s ) ) ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__DISJ__DIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1Q @ V3i ) ) @ ( ap @ V2R @ V3i ) ) ) ) )
<=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ V1Q @ V4i ) ) ) )
| ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ V2R @ V5i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__DISJ__EXISTS__DIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a )
@ ( lam @ A_27a
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V0P @ V3i ) ) @ ( ap @ V1Q @ V3i ) ) ) )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ V2R @ V4i ) ) ) )
<=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ V2R @ V5i ) ) ) )
| ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1Q )
@ ( lam @ A_27a
@ ^ [V6i: $i] : ( ap @ V2R @ V6i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__EQUAL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1j: $i] :
( ( mem @ V1j @ A_27a )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1j ) )
@ ( lam @ A_27a
@ ^ [V2i: $i] : ( ap @ V0P @ V2i ) ) ) )
<=> ( p @ ( ap @ V0P @ V1j ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__REORDER,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V3i: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27b ) @ V1Q )
@ ( lam @ A_27b
@ ^ [V4j: $i] : ( ap @ ( ap @ V2R @ V3i ) @ V4j ) ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27b ) @ V1Q )
@ ( lam @ A_27b
@ ^ [V5j: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V6i: $i] : ( ap @ ( ap @ V2R @ V6i ) @ V5j ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__F,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ A_27b )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27c @ bool ) )
=> ~ ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27c ) @ V2x )
@ ( lam @ A_27c
@ ^ [V3s: $i] : c_2Ebool_2EF ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__T,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27b ) @ V1s )
@ ( lam @ A_27b
@ ^ [V3x: $i] : c_2Ebool_2ET ) ) )
<=> ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0p ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIV,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0p ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V0p ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__NULL,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ bool )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V2x: $i] : V1m ) ) )
<=> ( ( V0p
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( p @ V1m ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__ALT,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0p ) @ V1m ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__SELECT @ A_27a ) @ V0p ) @ V1m ) ) @ V0p ) )
& ( p @ ( ap @ V1m @ ( ap @ ( ap @ ( c_2Ebool_2ERES__SELECT @ A_27a ) @ V0p ) @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ENOT__RES__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__NOT__EMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1s ) @ V0P ) )
=> ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1s ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V2t ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0P ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1s ) @ V0P ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V2t ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__DIFF,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0P @ V4x ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2t ) ) ) @ ( ap @ V0P @ V5x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2EIN__BIGUNION__RES__EXISTS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1sos ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ ( arr @ A_27a @ bool ) ) @ V1sos )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__BIGUNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1sos ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ ( arr @ A_27a @ bool ) ) @ V1sos )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3s: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V3s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0P @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__BIGINTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1sos ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ V1sos )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V4s: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V4s ) ) ) )
& ( p @ ( ap @ V0P @ V3x ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__ELIM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) )
<=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ V0P @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ V0P ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0P ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__F,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ~ ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27b ) @ V1s )
@ ( lam @ A_27b
@ ^ [V3x: $i] : c_2Ebool_2EF ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__T,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27b ) @ V1s )
@ ( lam @ A_27b
@ ^ [V3x: $i] : c_2Ebool_2ET ) ) )
<=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0p ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__NOT__EMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ V1s ) @ V0P ) )
=> ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__UNIV,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0p ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V0p ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__NULL,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ bool )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V2x: $i] : V1m ) ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( V0p
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
& ( p @ V1m ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__SING,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27b ) @ V1s )
@ ( lam @ A_27b
@ ^ [V3x: $i] : c_2Ebool_2ET ) ) )
<=> ? [V4y: $i] :
( ( mem @ V4y @ A_27b )
& ( V1s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V4y ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__EXISTS__UNIQUE__ALT,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ V0p ) @ V1m ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1m @ V2x ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V0p )
@ ( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1m @ V3y ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3y ) @ V2x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__SELECT__EMPTY,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ERES__SELECT @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0p )
= ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : c_2Ebool_2EF ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__SELECT__UNIV,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ERES__SELECT @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0p )
= ( ap @ ( c_2Emin_2E_40 @ A_27a ) @ V0p ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__ABSTRACT,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 ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__ABSTRACT__EQUAL,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0p ) )
=> ( ( ap @ V1m1 @ V3x )
= ( ap @ V2m2 @ V3x ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ V1m1 )
= ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ V2m2 ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__ABSTRACT__IDEMPOT,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ V1m ) )
= ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ V1m ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__ABSTRACT__EQUAL__EQ,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ V1m1 )
= ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0p ) @ V2m2 ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0p ) )
=> ( ( ap @ V1m1 @ V3x )
= ( ap @ V2m2 @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eres__quan_2ERES__ABSTRACT__UNIV,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0m )
= V0m ) ) ).
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