ITP001 Axioms: ITP032^5.ax


%------------------------------------------------------------------------------
% File     : ITP032^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

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

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