ITP001 Axioms: ITP003^7.ax


%------------------------------------------------------------------------------
% File     : ITP003^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : bool.ax [Gau19]
%          : HL4003^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  223 (  70 unt;  32 typ;   0 def)
%            Number of atoms       :  387 ( 140 equ;  45 cnn)
%            Maximal formula atoms :    8 (   1 avg)
%            Number of connectives : 1278 (  45   ~;  61   |; 126   &; 779   @)
%                                         ( 118 <=>; 149  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   6 avg; 779 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :  210 ( 210   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   32 (  30 usr;   3 con; 0-5 aty)
%            Number of variables   :  751 (  90   ^ 577   !;  56   ?; 751   :)
%                                         (  28  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ebool_2Eitself,type,
    tyop_2Ebool_2Eitself: $tType > $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(tyop_2Emin_2Eind,type,
    tyop_2Emin_2Eind: $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2E_3F_21,type,
    c_2Ebool_2E_3F_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Emin_2E_40,type,
    c_2Emin_2E_40: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).

thf(c_2Ebool_2EARB,type,
    c_2Ebool_2EARB: 
      !>[A_27a: $tType] : A_27a ).

thf(c_2Ebool_2EBOUNDED,type,
    c_2Ebool_2EBOUNDED: $o > $o ).

thf(c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Ebool_2EDATATYPE,type,
    c_2Ebool_2EDATATYPE: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Ebool_2EF,type,
    c_2Ebool_2EF: $o ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2ELET,type,
    c_2Ebool_2ELET: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).

thf(c_2Ebool_2EONE__ONE,type,
    c_2Ebool_2EONE__ONE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).

thf(c_2Ebool_2EONTO,type,
    c_2Ebool_2EONTO: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).

thf(c_2Ebool_2ERES__ABSTRACT,type,
    c_2Ebool_2ERES__ABSTRACT: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27b ) > A_27a > A_27b ) ).

thf(c_2Ebool_2ERES__EXISTS,type,
    c_2Ebool_2ERES__EXISTS: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2ERES__EXISTS__UNIQUE,type,
    c_2Ebool_2ERES__EXISTS__UNIQUE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2ERES__FORALL,type,
    c_2Ebool_2ERES__FORALL: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2ERES__SELECT,type,
    c_2Ebool_2ERES__SELECT: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a ) ).

thf(c_2Ebool_2ET,type,
    c_2Ebool_2ET: $o ).

thf(c_2Ebool_2ETYPE__DEFINITION,type,
    c_2Ebool_2ETYPE__DEFINITION: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Ebool_2Eitself__case,type,
    c_2Ebool_2Eitself__case: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ebool_2Eitself @ A_27a ) > A_27b > A_27b ) ).

thf(c_2Ebool_2Eliteral__case,type,
    c_2Ebool_2Eliteral__case: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).

thf(c_2Ebool_2Ethe__value,type,
    c_2Ebool_2Ethe__value: 
      !>[A_27a: $tType] : ( tyop_2Ebool_2Eitself @ A_27a ) ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Ebool_2EBOOL__CASES__AX,axiom,
    ! [V0t: $o] :
      ( ( V0t = c_2Ebool_2ET )
      | ( V0t = c_2Ebool_2EF ) ) ).

thf(thm_2Ebool_2EETA__AX,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0t: A_27a > A_27b] :
      ( ( ^ [V1x: A_27a] : ( V0t @ V1x ) )
      = V0t ) ).

thf(thm_2Ebool_2ESELECT__AX,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1x: A_27a] :
      ( ( V0P @ V1x )
     => ( V0P @ ( c_2Emin_2E_40 @ A_27a @ V0P ) ) ) ).

thf(thm_2Ebool_2EINFINITY__AX,axiom,
    ? [V0f: tyop_2Emin_2Eind > tyop_2Emin_2Eind] :
      ( ( c_2Ebool_2EONE__ONE @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ V0f )
      & ( (~) @ ( c_2Ebool_2EONTO @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ V0f ) ) ) ).

thf(thm_2Ebool_2ET__DEF,axiom,
    ( c_2Ebool_2ET
  <=> ( ( ^ [V0x: $o] : V0x )
      = ( ^ [V1x: $o] : V1x ) ) ) ).

thf(thm_2Ebool_2EFORALL__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2E_21 @ A_27a )
      = ( ^ [V0P: A_27a > $o] :
            ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V0P
            @ ^ [V1x: A_27a] : c_2Ebool_2ET ) ) ) ).

thf(thm_2Ebool_2EEXISTS__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2E_3F @ A_27a )
      = ( ^ [V0P: A_27a > $o] : ( V0P @ ( c_2Emin_2E_40 @ A_27a @ V0P ) ) ) ) ).

thf(thm_2Ebool_2EAND__DEF,axiom,
    ( c_2Ebool_2E_2F_5C
    = ( ^ [V0t1: $o,V1t2: $o] :
          ( c_2Ebool_2E_21 @ $o
          @ ^ [V2t: $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D_3D_3E @ V0t1 @ ( c_2Emin_2E_3D_3D_3E @ V1t2 @ V2t ) ) @ V2t ) ) ) ) ).

thf(thm_2Ebool_2EOR__DEF,axiom,
    ( c_2Ebool_2E_5C_2F
    = ( ^ [V0t1: $o,V1t2: $o] :
          ( c_2Ebool_2E_21 @ $o
          @ ^ [V2t: $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D_3D_3E @ V0t1 @ V2t ) @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D_3D_3E @ V1t2 @ V2t ) @ V2t ) ) ) ) ) ).

thf(thm_2Ebool_2EF__DEF,axiom,
    ( c_2Ebool_2EF
  <=> ! [V0t: $o] : V0t ) ).

thf(thm_2Ebool_2ENOT__DEF,axiom,
    ( c_2Ebool_2E_7E
    = ( ^ [V0t: $o] : ( c_2Emin_2E_3D_3D_3E @ V0t @ c_2Ebool_2EF ) ) ) ).

thf(thm_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2E_3F_21 @ A_27a )
      = ( ^ [V0P: A_27a > $o] :
            ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_3F @ A_27a @ V0P )
            @ ( c_2Ebool_2E_21 @ A_27a
              @ ^ [V1x: A_27a] :
                  ( c_2Ebool_2E_21 @ A_27a
                  @ ^ [V2y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V0P @ V1x ) @ ( V0P @ V2y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V2y ) ) ) ) ) ) ) ).

thf(thm_2Ebool_2ELET__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Ebool_2ELET @ A_27a @ A_27b )
      = ( ^ [V0f: A_27a > A_27b,V1x: A_27a] : ( V0f @ V1x ) ) ) ).

thf(thm_2Ebool_2ECOND__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2ECOND @ A_27a )
      = ( ^ [V0t: $o,V1t1: A_27a,V2t2: A_27a] :
            ( c_2Emin_2E_40 @ A_27a
            @ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ $o @ V0t @ c_2Ebool_2ET ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V1t1 ) ) @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ $o @ V0t @ c_2Ebool_2EF ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2t2 ) ) ) ) ) ) ).

thf(thm_2Ebool_2EONE__ONE__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Ebool_2EONE__ONE @ A_27a @ A_27b )
      = ( ^ [V0f: A_27a > A_27b] :
            ( c_2Ebool_2E_21 @ A_27a
            @ ^ [V1x1: A_27a] :
                ( c_2Ebool_2E_21 @ A_27a
                @ ^ [V2x2: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V1x1 ) @ ( V0f @ V2x2 ) ) @ ( c_2Emin_2E_3D @ A_27a @ V1x1 @ V2x2 ) ) ) ) ) ) ).

thf(thm_2Ebool_2EONTO__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Ebool_2EONTO @ A_27a @ A_27b )
      = ( ^ [V0f: A_27a > A_27b] :
            ( c_2Ebool_2E_21 @ A_27b
            @ ^ [V1y: A_27b] :
                ( c_2Ebool_2E_3F @ A_27a
                @ ^ [V2x: A_27a] : ( c_2Emin_2E_3D @ A_27b @ V1y @ ( V0f @ V2x ) ) ) ) ) ) ).

thf(thm_2Ebool_2ETYPE__DEFINITION,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b )
      = ( ^ [V0P: A_27a > $o,V1rep: A_27b > A_27a] :
            ( c_2Ebool_2E_2F_5C
            @ ( c_2Ebool_2E_21 @ A_27b
              @ ^ [V2x_27: A_27b] :
                  ( c_2Ebool_2E_21 @ A_27b
                  @ ^ [V3x_27_27: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ A_27a @ ( V1rep @ V2x_27 ) @ ( V1rep @ V3x_27_27 ) ) @ ( c_2Emin_2E_3D @ A_27b @ V2x_27 @ V3x_27_27 ) ) ) )
            @ ( c_2Ebool_2E_21 @ A_27a
              @ ^ [V4x: A_27a] :
                  ( c_2Emin_2E_3D @ $o @ ( V0P @ V4x )
                  @ ( c_2Ebool_2E_3F @ A_27b
                    @ ^ [V5x_27: A_27b] : ( c_2Emin_2E_3D @ A_27a @ V4x @ ( V1rep @ V5x_27 ) ) ) ) ) ) ) ) ).

thf(thm_2Ebool_2Eliteral__case__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b )
      = ( ^ [V0f: A_27a > A_27b,V1x: A_27a] : ( V0f @ V1x ) ) ) ).

thf(thm_2Ebool_2EIN__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2EIN @ A_27a )
      = ( ^ [V0x: A_27a,V1f: A_27a > $o] : ( V1f @ V0x ) ) ) ).

thf(thm_2Ebool_2ERES__FORALL__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2ERES__FORALL @ A_27a )
      = ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
            ( c_2Ebool_2E_21 @ A_27a
            @ ^ [V2x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p ) @ ( V1m @ V2x ) ) ) ) ) ).

thf(thm_2Ebool_2ERES__EXISTS__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2ERES__EXISTS @ A_27a )
      = ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
            ( c_2Ebool_2E_3F @ A_27a
            @ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p ) @ ( V1m @ V2x ) ) ) ) ) ).

thf(thm_2Ebool_2ERES__EXISTS__UNIQUE__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a )
      = ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
            ( c_2Ebool_2E_2F_5C
            @ ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0p
              @ ^ [V2x: A_27a] : ( V1m @ V2x ) )
            @ ( c_2Ebool_2ERES__FORALL @ A_27a @ V0p
              @ ^ [V3x: A_27a] :
                  ( c_2Ebool_2ERES__FORALL @ A_27a @ V0p
                  @ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1m @ V3x ) @ ( V1m @ V4y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V4y ) ) ) ) ) ) ) ).

thf(thm_2Ebool_2ERES__SELECT__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2ERES__SELECT @ A_27a )
      = ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
            ( c_2Emin_2E_40 @ A_27a
            @ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p ) @ ( V1m @ V2x ) ) ) ) ) ).

thf(thm_2Ebool_2EBOUNDED__DEF,axiom,
    ( c_2Ebool_2EBOUNDED
    = ( ^ [V0v: $o] : c_2Ebool_2ET ) ) ).

thf(thm_2Ebool_2EDATATYPE__TAG__DEF,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebool_2EDATATYPE @ A_27a )
      = ( ^ [V0x: A_27a] : c_2Ebool_2ET ) ) ).

thf(thm_2Ebool_2ERES__ABSTRACT__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0p: A_27a > $o,V1m: A_27a > A_27b,V2x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p )
         => ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m @ V2x )
            = ( V1m @ V2x ) ) )
      & ! [V3p: A_27a > $o,V4m1: A_27a > A_27b,V5m2: A_27a > A_27b] :
          ( ! [V6x: A_27a] :
              ( ( c_2Ebool_2EIN @ A_27a @ V6x @ V3p )
             => ( ( V4m1 @ V6x )
                = ( V5m2 @ V6x ) ) )
         => ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V3p @ V4m1 )
            = ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V3p @ V5m2 ) ) ) ) ).

thf(thm_2Ebool_2Eitself__TY__DEF,axiom,
    ! [A_27a: $tType] :
    ? [V0rep: ( tyop_2Ebool_2Eitself @ A_27a ) > A_27a] : ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ ( tyop_2Ebool_2Eitself @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Ebool_2EARB @ A_27a ) ) @ V0rep ) ).

thf(thm_2Ebool_2Eitself__case__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: A_27b] :
      ( ( c_2Ebool_2Eitself__case @ A_27a @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27a ) @ V0b )
      = V0b ) ).

thf(thm_2Ebool_2ETRUTH,axiom,
    c_2Ebool_2ET ).

thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1
       => V1t2 )
     => ( ( V1t2
         => V0t1 )
       => ( V0t1 = V1t2 ) ) ) ).

thf(thm_2Ebool_2EFALSITY,axiom,
    ! [V0t: $o] :
      ( c_2Ebool_2EF
     => V0t ) ).

thf(thm_2Ebool_2EETA__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0M: A_27a > A_27b] :
      ( ( ^ [V1x: A_27a] : ( V0M @ V1x ) )
      = V0M ) ).

thf(thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
    ! [V0t: $o] :
      ( V0t
      | ( (~) @ V0t ) ) ).

thf(thm_2Ebool_2EBETA__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1y: A_27a] :
      ( ( ^ [V2x: A_27a] : ( V0f @ V2x )
        @ V1y )
      = ( V0f @ V1y ) ) ).

thf(thm_2Ebool_2ELET__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a] :
      ( ( c_2Ebool_2ELET @ A_27a @ A_27b @ V0f @ V1x )
      = ( V0f @ V1x ) ) ).

thf(thm_2Ebool_2EFORALL__THM,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Ebool_2EEXISTS__THM,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Ebool_2EABS__SIMP,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0t1: A_27a,V1t2: A_27b] :
      ( ( ^ [V2x: A_27b] : V0t1
        @ V1t2 )
      = V0t1 ) ).

thf(thm_2Ebool_2EFORALL__SIMP,axiom,
    ! [A_27a: $tType,V0t: $o] :
      ( ! [V1x: A_27a] : V0t
    <=> V0t ) ).

thf(thm_2Ebool_2EEXISTS__SIMP,axiom,
    ! [A_27a: $tType,V0t: $o] :
      ( ? [V1x: A_27a] : V0t
    <=> V0t ) ).

thf(thm_2Ebool_2EAND__INTRO__THM,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( V0t1
     => ( V1t2
       => ( V0t1
          & V1t2 ) ) ) ).

thf(thm_2Ebool_2EAND1__THM,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1
        & V1t2 )
     => V0t1 ) ).

thf(thm_2Ebool_2EAND2__THM,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1
        & V1t2 )
     => V1t2 ) ).

thf(thm_2Ebool_2ECONJ__SYM,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1
        & V1t2 )
    <=> ( V1t2
        & V0t1 ) ) ).

thf(thm_2Ebool_2ECONJ__COMM,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1
        & V1t2 )
    <=> ( V1t2
        & V0t1 ) ) ).

thf(thm_2Ebool_2ECONJ__ASSOC,axiom,
    ! [V0t1: $o,V1t2: $o,V2t3: $o] :
      ( ( V0t1
        & V1t2
        & V2t3 )
    <=> ( V0t1
        & V1t2
        & V2t3 ) ) ).

thf(thm_2Ebool_2EOR__INTRO__THM1,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( V0t1
     => ( V0t1
        | V1t2 ) ) ).

thf(thm_2Ebool_2EOR__INTRO__THM2,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( V1t2
     => ( V0t1
        | V1t2 ) ) ).

thf(thm_2Ebool_2EOR__ELIM__THM,axiom,
    ! [V0t: $o,V1t1: $o,V2t2: $o] :
      ( ( V1t1
        | V2t2 )
     => ( ( V1t1
         => V0t )
       => ( ( V2t2
           => V0t )
         => V0t ) ) ) ).

thf(thm_2Ebool_2EIMP__F,axiom,
    ! [V0t: $o] :
      ( ( V0t
       => c_2Ebool_2EF )
     => ( (~) @ V0t ) ) ).

thf(thm_2Ebool_2EF__IMP,axiom,
    ! [V0t: $o] :
      ( ( (~) @ V0t )
     => ( V0t
       => c_2Ebool_2EF ) ) ).

thf(thm_2Ebool_2ENOT__F,axiom,
    ! [V0t: $o] :
      ( ( (~) @ V0t )
     => ( V0t = c_2Ebool_2EF ) ) ).

thf(thm_2Ebool_2ENOT__AND,axiom,
    ! [V0t: $o] :
      ( (~)
      @ ( V0t
        & ( (~) @ V0t ) ) ) ).

thf(thm_2Ebool_2EAND__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET
          & V0t )
      <=> V0t )
      & ( ( V0t
          & c_2Ebool_2ET )
      <=> V0t )
      & ( ( c_2Ebool_2EF
          & V0t )
      <=> c_2Ebool_2EF )
      & ( ( V0t
          & c_2Ebool_2EF )
      <=> c_2Ebool_2EF )
      & ( ( V0t
          & V0t )
      <=> V0t ) ) ).

thf(thm_2Ebool_2EOR__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET
          | V0t )
      <=> c_2Ebool_2ET )
      & ( ( V0t
          | c_2Ebool_2ET )
      <=> c_2Ebool_2ET )
      & ( ( c_2Ebool_2EF
          | V0t )
      <=> V0t )
      & ( ( V0t
          | c_2Ebool_2EF )
      <=> V0t )
      & ( ( V0t
          | V0t )
      <=> V0t ) ) ).

thf(thm_2Ebool_2EIMP__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET
         => V0t )
      <=> V0t )
      & ( ( V0t
         => c_2Ebool_2ET )
      <=> c_2Ebool_2ET )
      & ( ( c_2Ebool_2EF
         => V0t )
      <=> c_2Ebool_2ET )
      & ( ( V0t
         => V0t )
      <=> c_2Ebool_2ET )
      & ( ( V0t
         => c_2Ebool_2EF )
      <=> ( (~) @ V0t ) ) ) ).

thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
    ( ! [V0t: $o] :
        ( ( (~) @ ( (~) @ V0t ) )
      <=> V0t )
    & ( ( (~) @ c_2Ebool_2ET )
    <=> c_2Ebool_2EF )
    & ( ( (~) @ c_2Ebool_2EF )
    <=> c_2Ebool_2ET ) ) ).

thf(thm_2Ebool_2EEQ__REFL,axiom,
    ! [A_27a: $tType,V0x: A_27a] : ( V0x = V0x ) ).

thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( V0x = V0x )
    <=> c_2Ebool_2ET ) ).

thf(thm_2Ebool_2EEQ__SYM,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
      ( ( V0x = V1y )
     => ( V1y = V0x ) ) ).

thf(thm_2Ebool_2EEQ__SYM__EQ,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
      ( ( V0x = V1y )
    <=> ( V1y = V0x ) ) ).

thf(thm_2Ebool_2EEQ__EXT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b] :
      ( ! [V2x: A_27a] :
          ( ( V0f @ V2x )
          = ( V1g @ V2x ) )
     => ( V0f = V1g ) ) ).

thf(thm_2Ebool_2EFUN__EQ__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b] :
      ( ( V0f = V1g )
    <=> ! [V2x: A_27a] :
          ( ( V0f @ V2x )
          = ( V1g @ V2x ) ) ) ).

thf(thm_2Ebool_2EEQ__TRANS,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2z: A_27a] :
      ( ( ( V0x = V1y )
        & ( V1y = V2z ) )
     => ( V0x = V2z ) ) ).

thf(thm_2Ebool_2EBOOL__EQ__DISTINCT,axiom,
    ( ( (~) @ ( c_2Ebool_2ET = c_2Ebool_2EF ) )
    & ( (~) @ ( c_2Ebool_2EF = c_2Ebool_2ET ) ) ) ).

thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET = V0t )
      <=> V0t )
      & ( ( V0t = c_2Ebool_2ET )
      <=> V0t )
      & ( ( c_2Ebool_2EF = V0t )
      <=> ( (~) @ V0t ) )
      & ( ( V0t = c_2Ebool_2EF )
      <=> ( (~) @ V0t ) ) ) ).

thf(thm_2Ebool_2ECOND__CLAUSES,axiom,
    ! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
      ( ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
        = V0t1 )
      & ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V0t1 @ V1t2 )
        = V1t2 ) ) ).

thf(thm_2Ebool_2ECOND__ID,axiom,
    ! [A_27a: $tType,V0b: $o,V1t: A_27a] :
      ( ( c_2Ebool_2ECOND @ A_27a @ V0b @ V1t @ V1t )
      = V1t ) ).

thf(thm_2Ebool_2ESELECT__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( V0P
        @ ( c_2Emin_2E_40 @ A_27a
          @ ^ [V1x: A_27a] : ( V0P @ V1x ) ) )
    <=> ? [V2x: A_27a] : ( V0P @ V2x ) ) ).

thf(thm_2Ebool_2ESELECT__REFL,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( c_2Emin_2E_40 @ A_27a
        @ ^ [V1y: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V1y @ V0x ) )
      = V0x ) ).

thf(thm_2Ebool_2ESELECT__REFL__2,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( c_2Emin_2E_40 @ A_27a
        @ ^ [V1y: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V0x @ V1y ) )
      = V0x ) ).

thf(thm_2Ebool_2ESELECT__UNIQUE,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1x: A_27a] :
      ( ! [V2y: A_27a] :
          ( ( V0P @ V2y )
        <=> ( V2y = V1x ) )
     => ( ( c_2Emin_2E_40 @ A_27a @ V0P )
        = V1x ) ) ).

thf(thm_2Ebool_2ESELECT__ELIM__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
      ( ( ? [V2x: A_27a] : ( V0P @ V2x )
        & ! [V3x: A_27a] :
            ( ( V0P @ V3x )
           => ( V1Q @ V3x ) ) )
     => ( V1Q @ ( c_2Emin_2E_40 @ A_27a @ V0P ) ) ) ).

thf(thm_2Ebool_2ENOT__FORALL__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( (~)
        @ ! [V1x: A_27a] : ( V0P @ V1x ) )
    <=> ? [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).

thf(thm_2Ebool_2ENOT__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( (~)
        @ ? [V1x: A_27a] : ( V0P @ V1x ) )
    <=> ! [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).

thf(thm_2Ebool_2EFORALL__AND__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( ( V0P @ V2x )
          & ( V1Q @ V2x ) )
    <=> ( ! [V3x: A_27a] : ( V0P @ V3x )
        & ! [V4x: A_27a] : ( V1Q @ V4x ) ) ) ).

thf(thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ( ! [V2x: A_27a] : ( V0P @ V2x )
        & V1Q )
    <=> ! [V3x: A_27a] :
          ( ( V0P @ V3x )
          & V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ( V0P
        & ! [V2x: A_27a] : ( V1Q @ V2x ) )
    <=> ! [V3x: A_27a] :
          ( V0P
          & ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EEXISTS__OR__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
      ( ? [V2x: A_27a] :
          ( ( V0P @ V2x )
          | ( V1Q @ V2x ) )
    <=> ( ? [V3x: A_27a] : ( V0P @ V3x )
        | ? [V4x: A_27a] : ( V1Q @ V4x ) ) ) ).

thf(thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ( ? [V2x: A_27a] : ( V0P @ V2x )
        | V1Q )
    <=> ? [V3x: A_27a] :
          ( ( V0P @ V3x )
          | V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ( V0P
        | ? [V2x: A_27a] : ( V1Q @ V2x ) )
    <=> ? [V3x: A_27a] :
          ( V0P
          | ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EBOTH__EXISTS__AND__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: $o] :
      ( ? [V2x: A_27a] :
          ( V0P
          & V1Q )
    <=> ( ? [V3x: A_27a] : V0P
        & ? [V4x: A_27a] : V1Q ) ) ).

thf(thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ? [V2x: A_27a] :
          ( ( V0P @ V2x )
          & V1Q )
    <=> ( ? [V3x: A_27a] : ( V0P @ V3x )
        & V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ? [V2x: A_27a] :
          ( V0P
          & ( V1Q @ V2x ) )
    <=> ( V0P
        & ? [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EBOTH__FORALL__OR__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: $o] :
      ( ! [V2x: A_27a] :
          ( V0P
          | V1Q )
    <=> ( ! [V3x: A_27a] : V0P
        | ! [V4x: A_27a] : V1Q ) ) ).

thf(thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
    ! [A_27a: $tType,V0Q: $o,V1P: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( ( V1P @ V2x )
          | V0Q )
    <=> ( ! [V3x: A_27a] : ( V1P @ V3x )
        | V0Q ) ) ).

thf(thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( V0P
          | ( V1Q @ V2x ) )
    <=> ( V0P
        | ! [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EBOTH__FORALL__IMP__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: $o] :
      ( ! [V2x: A_27a] :
          ( V0P
         => V1Q )
    <=> ( ? [V3x: A_27a] : V0P
       => ! [V4x: A_27a] : V1Q ) ) ).

thf(thm_2Ebool_2ELEFT__FORALL__IMP__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ! [V2x: A_27a] :
          ( ( V0P @ V2x )
         => V1Q )
    <=> ( ? [V3x: A_27a] : ( V0P @ V3x )
       => V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__FORALL__IMP__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( V0P
         => ( V1Q @ V2x ) )
    <=> ( V0P
       => ! [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EBOTH__EXISTS__IMP__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: $o] :
      ( ? [V2x: A_27a] :
          ( V0P
         => V1Q )
    <=> ( ! [V3x: A_27a] : V0P
       => ? [V4x: A_27a] : V1Q ) ) ).

thf(thm_2Ebool_2ELEFT__EXISTS__IMP__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ? [V2x: A_27a] :
          ( ( V0P @ V2x )
         => V1Q )
    <=> ( ! [V3x: A_27a] : ( V0P @ V3x )
       => V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__EXISTS__IMP__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ? [V2x: A_27a] :
          ( V0P
         => ( V1Q @ V2x ) )
    <=> ( V0P
       => ? [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EOR__IMP__THM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( V0A
      <=> ( V1B
          | V0A ) )
    <=> ( V1B
       => V0A ) ) ).

thf(thm_2Ebool_2ENOT__IMP,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( (~)
        @ ( V0A
         => V1B ) )
    <=> ( V0A
        & ( (~) @ V1B ) ) ) ).

thf(thm_2Ebool_2EDISJ__ASSOC,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( V0A
        | V1B
        | V2C )
    <=> ( V0A
        | V1B
        | V2C ) ) ).

thf(thm_2Ebool_2EDISJ__SYM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( V0A
        | V1B )
    <=> ( V1B
        | V0A ) ) ).

thf(thm_2Ebool_2EDISJ__COMM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( V0A
        | V1B )
    <=> ( V1B
        | V0A ) ) ).

thf(thm_2Ebool_2EDE__MORGAN__THM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( ( (~)
          @ ( V0A
            & V1B ) )
      <=> ( ( (~) @ V0A )
          | ( (~) @ V1B ) ) )
      & ( ( (~)
          @ ( V0A
            | V1B ) )
      <=> ( ( (~) @ V0A )
          & ( (~) @ V1B ) ) ) ) ).

thf(thm_2Ebool_2ELEFT__AND__OVER__OR,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( V0A
        & ( V1B
          | V2C ) )
    <=> ( ( V0A
          & V1B )
        | ( V0A
          & V2C ) ) ) ).

thf(thm_2Ebool_2ERIGHT__AND__OVER__OR,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( ( V1B
          | V2C )
        & V0A )
    <=> ( ( V1B
          & V0A )
        | ( V2C
          & V0A ) ) ) ).

thf(thm_2Ebool_2ELEFT__OR__OVER__AND,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( V0A
        | ( V1B
          & V2C ) )
    <=> ( ( V0A
          | V1B )
        & ( V0A
          | V2C ) ) ) ).

thf(thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( ( V1B
          & V2C )
        | V0A )
    <=> ( ( V1B
          | V0A )
        & ( V2C
          | V0A ) ) ) ).

thf(thm_2Ebool_2EIMP__DISJ__THM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( V0A
       => V1B )
    <=> ( ( (~) @ V0A )
        | V1B ) ) ).

thf(thm_2Ebool_2EDISJ__IMP__THM,axiom,
    ! [V0P: $o,V1Q: $o,V2R: $o] :
      ( ( ( V0P
          | V1Q )
       => V2R )
    <=> ( ( V0P
         => V2R )
        & ( V1Q
         => V2R ) ) ) ).

thf(thm_2Ebool_2EIMP__CONJ__THM,axiom,
    ! [V0P: $o,V1Q: $o,V2R: $o] :
      ( ( V0P
       => ( V1Q
          & V2R ) )
    <=> ( ( V0P
         => V1Q )
        & ( V0P
         => V2R ) ) ) ).

thf(thm_2Ebool_2EIMP__F__EQ__F,axiom,
    ! [V0t: $o] :
      ( ( V0t
       => c_2Ebool_2EF )
    <=> ( V0t = c_2Ebool_2EF ) ) ).

thf(thm_2Ebool_2EAND__IMP__INTRO,axiom,
    ! [V0t1: $o,V1t2: $o,V2t3: $o] :
      ( ( V0t1
       => ( V1t2
         => V2t3 ) )
    <=> ( ( V0t1
          & V1t2 )
       => V2t3 ) ) ).

thf(thm_2Ebool_2EEQ__IMP__THM,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1 = V1t2 )
    <=> ( ( V0t1
         => V1t2 )
        & ( V1t2
         => V0t1 ) ) ) ).

thf(thm_2Ebool_2EEQ__EXPAND,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1 = V1t2 )
    <=> ( ( V0t1
          & V1t2 )
        | ( ( (~) @ V0t1 )
          & ( (~) @ V1t2 ) ) ) ) ).

thf(thm_2Ebool_2ECOND__RATOR,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: $o,V1f: A_27a > A_27b,V2g: A_27a > A_27b,V3x: A_27a] :
      ( ( c_2Ebool_2ECOND @ ( A_27a > A_27b ) @ V0b @ V1f @ V2g @ V3x )
      = ( c_2Ebool_2ECOND @ A_27b @ V0b @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) ) ).

thf(thm_2Ebool_2ECOND__RAND,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1b: $o,V2x: A_27a,V3y: A_27a] :
      ( ( V0f @ ( c_2Ebool_2ECOND @ A_27a @ V1b @ V2x @ V3y ) )
      = ( c_2Ebool_2ECOND @ A_27b @ V1b @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) ) ).

thf(thm_2Ebool_2ECOND__ABS,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: $o,V1f: A_27a > A_27b,V2g: A_27a > A_27b] :
      ( ( ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ V0b @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) )
      = ( c_2Ebool_2ECOND @ ( A_27a > A_27b ) @ V0b @ V1f @ V2g ) ) ).

thf(thm_2Ebool_2ECOND__EXPAND,axiom,
    ! [V0b: $o,V1t1: $o,V2t2: $o] :
      ( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
    <=> ( ( ( (~) @ V0b )
          | V1t1 )
        & ( V0b
          | V2t2 ) ) ) ).

thf(thm_2Ebool_2ECOND__EXPAND__IMP,axiom,
    ! [V0b: $o,V1t1: $o,V2t2: $o] :
      ( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
    <=> ( ( V0b
         => V1t1 )
        & ( ( (~) @ V0b )
         => V2t2 ) ) ) ).

thf(thm_2Ebool_2ECOND__EXPAND__OR,axiom,
    ! [V0b: $o,V1t1: $o,V2t2: $o] :
      ( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
    <=> ( ( V0b
          & V1t1 )
        | ( ( (~) @ V0b )
          & V2t2 ) ) ) ).

thf(thm_2Ebool_2ETYPE__DEFINITION__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1rep: A_27b > A_27a] :
      ( ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b @ V0P @ V1rep )
    <=> ( ! [V2x_27: A_27b,V3x_27_27: A_27b] :
            ( ( ( V1rep @ V2x_27 )
              = ( V1rep @ V3x_27_27 ) )
           => ( V2x_27 = V3x_27_27 ) )
        & ! [V4x: A_27a] :
            ( ( V0P @ V4x )
          <=> ? [V5x_27: A_27b] :
                ( V4x
                = ( V1rep @ V5x_27 ) ) ) ) ) ).

thf(thm_2Ebool_2EONTO__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
      ( ( c_2Ebool_2EONTO @ A_27a @ A_27b @ V0f )
    <=> ! [V1y: A_27b] :
        ? [V2x: A_27a] :
          ( V1y
          = ( V0f @ V2x ) ) ) ).

thf(thm_2Ebool_2EONE__ONE__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
      ( ( c_2Ebool_2EONE__ONE @ A_27a @ A_27b @ V0f )
    <=> ! [V1x1: A_27a,V2x2: A_27a] :
          ( ( ( V0f @ V1x1 )
            = ( V0f @ V2x2 ) )
         => ( V1x1 = V2x2 ) ) ) ).

thf(thm_2Ebool_2EABS__REP__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o] :
      ( ? [V1rep: A_27b > A_27a] : ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b @ V0P @ V1rep )
     => ? [V2rep: A_27b > A_27a,V3abs: A_27a > A_27b] :
          ( ! [V4a: A_27b] :
              ( ( V3abs @ ( V2rep @ V4a ) )
              = V4a )
          & ! [V5r: A_27a] :
              ( ( V0P @ V5r )
            <=> ( ( V2rep @ ( V3abs @ V5r ) )
                = V5r ) ) ) ) ).

thf(thm_2Ebool_2ELET__RAND,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27b > $o,V1N: A_27a > A_27b,V2M: A_27a] :
      ( ( V0P
        @ ( c_2Ebool_2ELET @ A_27a @ A_27b
          @ ^ [V3x: A_27a] : ( V1N @ V3x )
          @ V2M ) )
      = ( c_2Ebool_2ELET @ A_27a @ $o
        @ ^ [V4x: A_27a] : ( V0P @ ( V1N @ V4x ) )
        @ V2M ) ) ).

thf(thm_2Ebool_2ELET__RATOR,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0b: A_27b,V1N: A_27a > A_27b > A_27c,V2M: A_27a] :
      ( ( c_2Ebool_2ELET @ A_27a @ ( A_27b > A_27c )
        @ ^ [V3x: A_27a] : ( V1N @ V3x )
        @ V2M
        @ V0b )
      = ( c_2Ebool_2ELET @ A_27a @ A_27c
        @ ^ [V4x: A_27a] : ( V1N @ V4x @ V0b )
        @ V2M ) ) ).

thf(thm_2Ebool_2ESWAP__FORALL__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
      ( ! [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y )
    <=> ! [V3y: A_27b,V4x: A_27a] : ( V0P @ V4x @ V3y ) ) ).

thf(thm_2Ebool_2ESWAP__EXISTS__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
      ( ? [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y )
    <=> ? [V3y: A_27b,V4x: A_27a] : ( V0P @ V4x @ V3y ) ) ).

thf(thm_2Ebool_2EEXISTS__UNIQUE__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( c_2Ebool_2E_3F_21 @ A_27a
        @ ^ [V1x: A_27a] : ( V0P @ V1x ) )
    <=> ( ? [V2x: A_27a] : ( V0P @ V2x )
        & ! [V3x: A_27a,V4y: A_27a] :
            ( ( ( V0P @ V3x )
              & ( V0P @ V4y ) )
           => ( V3x = V4y ) ) ) ) ).

thf(thm_2Ebool_2ELET__CONG,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b,V2M: A_27a,V3N: A_27a] :
      ( ( ( V2M = V3N )
        & ! [V4x: A_27a] :
            ( ( V4x = V3N )
           => ( ( V0f @ V4x )
              = ( V1g @ V4x ) ) ) )
     => ( ( c_2Ebool_2ELET @ A_27a @ A_27b @ V0f @ V2M )
        = ( c_2Ebool_2ELET @ A_27a @ A_27b @ V1g @ V3N ) ) ) ).

thf(thm_2Ebool_2EIMP__CONG,axiom,
    ! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
      ( ( ( V0x = V1x_27 )
        & ( V1x_27
         => ( V2y = V3y_27 ) ) )
     => ( ( V0x
         => V2y )
      <=> ( V1x_27
         => V3y_27 ) ) ) ).

thf(thm_2Ebool_2EAND__CONG,axiom,
    ! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
      ( ( ( V2Q
         => ( V0P = V1P_27 ) )
        & ( V1P_27
         => ( V2Q = V3Q_27 ) ) )
     => ( ( V0P
          & V2Q )
      <=> ( V1P_27
          & V3Q_27 ) ) ) ).

thf(thm_2Ebool_2ELEFT__AND__CONG,axiom,
    ! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
      ( ( ( V0P = V1P_27 )
        & ( V1P_27
         => ( V2Q = V3Q_27 ) ) )
     => ( ( V0P
          & V2Q )
      <=> ( V1P_27
          & V3Q_27 ) ) ) ).

thf(thm_2Ebool_2EOR__CONG,axiom,
    ! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
      ( ( ( ( (~) @ V2Q )
         => ( V0P = V1P_27 ) )
        & ( ( (~) @ V1P_27 )
         => ( V2Q = V3Q_27 ) ) )
     => ( ( V0P
          | V2Q )
      <=> ( V1P_27
          | V3Q_27 ) ) ) ).

thf(thm_2Ebool_2ELEFT__OR__CONG,axiom,
    ! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
      ( ( ( V0P = V1P_27 )
        & ( ( (~) @ V1P_27 )
         => ( V2Q = V3Q_27 ) ) )
     => ( ( V0P
          | V2Q )
      <=> ( V1P_27
          | V3Q_27 ) ) ) ).

thf(thm_2Ebool_2ECOND__CONG,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: $o,V2x: A_27a,V3x_27: A_27a,V4y: A_27a,V5y_27: A_27a] :
      ( ( ( V0P = V1Q )
        & ( V1Q
         => ( V2x = V3x_27 ) )
        & ( ( (~) @ V1Q )
         => ( V4y = V5y_27 ) ) )
     => ( ( c_2Ebool_2ECOND @ A_27a @ V0P @ V2x @ V4y )
        = ( c_2Ebool_2ECOND @ A_27a @ V1Q @ V3x_27 @ V5y_27 ) ) ) ).

thf(thm_2Ebool_2ERES__FORALL__CONG,axiom,
    ! [A_27a: $tType,V0g: A_27a > $o,V1f: A_27a > $o,V2Q: A_27a > $o,V3P: A_27a > $o] :
      ( ( V3P = V2Q )
     => ( ! [V4x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2Q )
           => ( ( V1f @ V4x )
              = ( V0g @ V4x ) ) )
       => ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V3P @ V1f )
          = ( c_2Ebool_2ERES__FORALL @ A_27a @ V2Q @ V0g ) ) ) ) ).

thf(thm_2Ebool_2ERES__EXISTS__CONG,axiom,
    ! [A_27a: $tType,V0g: A_27a > $o,V1f: A_27a > $o,V2Q: A_27a > $o,V3P: A_27a > $o] :
      ( ( V3P = V2Q )
     => ( ! [V4x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2Q )
           => ( ( V1f @ V4x )
              = ( V0g @ V4x ) ) )
       => ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V3P @ V1f )
          = ( c_2Ebool_2ERES__EXISTS @ A_27a @ V2Q @ V0g ) ) ) ) ).

thf(thm_2Ebool_2EMONO__AND,axiom,
    ! [V0z: $o,V1y: $o,V2x: $o,V3w: $o] :
      ( ( ( V2x
         => V1y )
        & ( V0z
         => V3w ) )
     => ( ( V2x
          & V0z )
       => ( V1y
          & V3w ) ) ) ).

thf(thm_2Ebool_2EMONO__OR,axiom,
    ! [V0z: $o,V1y: $o,V2x: $o,V3w: $o] :
      ( ( ( V2x
         => V1y )
        & ( V0z
         => V3w ) )
     => ( ( V2x
          | V0z )
       => ( V1y
          | V3w ) ) ) ).

thf(thm_2Ebool_2EMONO__IMP,axiom,
    ! [V0z: $o,V1y: $o,V2x: $o,V3w: $o] :
      ( ( ( V1y
         => V2x )
        & ( V0z
         => V3w ) )
     => ( ( V2x
         => V0z )
       => ( V1y
         => V3w ) ) ) ).

thf(thm_2Ebool_2EMONO__NOT,axiom,
    ! [V0y: $o,V1x: $o] :
      ( ( V0y
       => V1x )
     => ( ( (~) @ V1x )
       => ( (~) @ V0y ) ) ) ).

thf(thm_2Ebool_2EMONO__NOT__EQ,axiom,
    ! [V0y: $o,V1x: $o] :
      ( ( V0y
       => V1x )
    <=> ( ( (~) @ V1x )
       => ( (~) @ V0y ) ) ) ).

thf(thm_2Ebool_2EMONO__ALL,axiom,
    ! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( ( V1P @ V2x )
         => ( V0Q @ V2x ) )
     => ( ! [V3x: A_27a] : ( V1P @ V3x )
       => ! [V4x: A_27a] : ( V0Q @ V4x ) ) ) ).

thf(thm_2Ebool_2EMONO__EXISTS,axiom,
    ! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( ( V1P @ V2x )
         => ( V0Q @ V2x ) )
     => ( ? [V3x: A_27a] : ( V1P @ V3x )
       => ? [V4x: A_27a] : ( V0Q @ V4x ) ) ) ).

thf(thm_2Ebool_2EMONO__COND,axiom,
    ! [V0z: $o,V1y: $o,V2x: $o,V3w: $o,V4b: $o] :
      ( ( V2x
       => V1y )
     => ( ( V0z
         => V3w )
       => ( ( c_2Ebool_2ECOND @ $o @ V4b @ V2x @ V0z )
         => ( c_2Ebool_2ECOND @ $o @ V4b @ V1y @ V3w ) ) ) ) ).

thf(thm_2Ebool_2EEXISTS__REFL,axiom,
    ! [A_27a: $tType,V0a: A_27a] :
    ? [V1x: A_27a] : ( V1x = V0a ) ).

thf(thm_2Ebool_2EEXISTS__UNIQUE__REFL,axiom,
    ! [A_27a: $tType,V0a: A_27a] :
      ( c_2Ebool_2E_3F_21 @ A_27a
      @ ^ [V1x: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V1x @ V0a ) ) ).

thf(thm_2Ebool_2EUNWIND__THM1,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a] :
      ( ? [V2x: A_27a] :
          ( ( V1a = V2x )
          & ( V0P @ V2x ) )
    <=> ( V0P @ V1a ) ) ).

thf(thm_2Ebool_2EUNWIND__THM2,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a] :
      ( ? [V2x: A_27a] :
          ( ( V2x = V1a )
          & ( V0P @ V2x ) )
    <=> ( V0P @ V1a ) ) ).

thf(thm_2Ebool_2EUNWIND__FORALL__THM1,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o,V1v: A_27a] :
      ( ! [V2x: A_27a] :
          ( ( V1v = V2x )
         => ( V0f @ V2x ) )
    <=> ( V0f @ V1v ) ) ).

thf(thm_2Ebool_2EUNWIND__FORALL__THM2,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o,V1v: A_27a] :
      ( ! [V2x: A_27a] :
          ( ( V2x = V1v )
         => ( V0f @ V2x ) )
    <=> ( V0f @ V1v ) ) ).

thf(thm_2Ebool_2ESKOLEM__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
      ( ! [V1x: A_27a] :
        ? [V2y: A_27b] : ( V0P @ V1x @ V2y )
    <=> ? [V3f: A_27a > A_27b] :
        ! [V4x: A_27a] : ( V0P @ V4x @ ( V3f @ V4x ) ) ) ).

thf(thm_2Ebool_2Ebool__case__thm,axiom,
    ! [A_27a: $tType] :
      ( ! [V0t1: A_27a,V1t2: A_27a] :
          ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
          = V0t1 )
      & ! [V2t1: A_27a,V3t2: A_27a] :
          ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V2t1 @ V3t2 )
          = V3t2 ) ) ).

thf(thm_2Ebool_2Ebool__case__ID,axiom,
    ! [A_27a: $tType,V0b: $o,V1t: A_27a] :
      ( ( c_2Ebool_2ECOND @ A_27a @ V0b @ V1t @ V1t )
      = V1t ) ).

thf(thm_2Ebool_2EboolAxiom,axiom,
    ! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
    ? [V2fn: $o > A_27a] :
      ( ( ( V2fn @ c_2Ebool_2ET )
        = V0t1 )
      & ( ( V2fn @ c_2Ebool_2EF )
        = V1t2 ) ) ).

thf(thm_2Ebool_2Ebool__INDUCT,axiom,
    ! [V0P: $o > $o] :
      ( ( ( V0P @ c_2Ebool_2ET )
        & ( V0P @ c_2Ebool_2EF ) )
     => ! [V1b: $o] : ( V0P @ V1b ) ) ).

thf(thm_2Ebool_2Ebool__case__CONG,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: $o,V2x: A_27a,V3x_27: A_27a,V4y: A_27a,V5y_27: A_27a] :
      ( ( ( V0P = V1Q )
        & ( V1Q
         => ( V2x = V3x_27 ) )
        & ( ( (~) @ V1Q )
         => ( V4y = V5y_27 ) ) )
     => ( ( c_2Ebool_2ECOND @ A_27a @ V0P @ V2x @ V4y )
        = ( c_2Ebool_2ECOND @ A_27a @ V1Q @ V3x_27 @ V5y_27 ) ) ) ).

thf(thm_2Ebool_2EFORALL__BOOL,axiom,
    ! [V0P: $o > $o] :
      ( ! [V1b: $o] : ( V0P @ V1b )
    <=> ( ( V0P @ c_2Ebool_2ET )
        & ( V0P @ c_2Ebool_2EF ) ) ) ).

thf(thm_2Ebool_2EUEXISTS__OR__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
      ( ( c_2Ebool_2E_3F_21 @ A_27a
        @ ^ [V2x: A_27a] : ( c_2Ebool_2E_5C_2F @ ( V0P @ V2x ) @ ( V1Q @ V2x ) ) )
     => ( ( c_2Ebool_2E_3F_21 @ A_27a
          @ ^ [V3x: A_27a] : ( V0P @ V3x ) )
        | ( c_2Ebool_2E_3F_21 @ A_27a
          @ ^ [V4x: A_27a] : ( V1Q @ V4x ) ) ) ) ).

thf(thm_2Ebool_2EUEXISTS__SIMP,axiom,
    ! [A_27a: $tType,V0t: $o] :
      ( ( c_2Ebool_2E_3F_21 @ A_27a
        @ ^ [V1x: A_27a] : V0t )
    <=> ( V0t
        & ! [V2x: A_27a,V3y: A_27a] : ( V2x = V3y ) ) ) ).

thf(thm_2Ebool_2ERES__FORALL__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
      ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P @ V1f )
    <=> ! [V2x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
         => ( V1f @ V2x ) ) ) ).

thf(thm_2Ebool_2ERES__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
      ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P @ V1f )
    <=> ? [V2x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
          & ( V1f @ V2x ) ) ) ).

thf(thm_2Ebool_2ERES__EXISTS__UNIQUE__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
      ( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V0P @ V1f )
    <=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
          @ ^ [V2x: A_27a] : ( V1f @ V2x ) )
        & ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
          @ ^ [V3x: A_27a] :
              ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
              @ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1f @ V3x ) @ ( V1f @ V4y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V4y ) ) ) ) ) ) ).

thf(thm_2Ebool_2ERES__SELECT__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
      ( ( c_2Ebool_2ERES__SELECT @ A_27a @ V0P @ V1f )
      = ( c_2Emin_2E_40 @ A_27a
        @ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P ) @ ( V1f @ V2x ) ) ) ) ).

thf(thm_2Ebool_2ERES__FORALL__TRUE,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
        @ ^ [V1x: A_27a] : c_2Ebool_2ET )
      = c_2Ebool_2ET ) ).

thf(thm_2Ebool_2ERES__EXISTS__FALSE,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
        @ ^ [V1x: A_27a] : c_2Ebool_2EF )
      = c_2Ebool_2EF ) ).

thf(thm_2Ebool_2EBOOL__FUN__CASES__THM,axiom,
    ! [V0f: $o > $o] :
      ( ( V0f
        = ( ^ [V1b: $o] : c_2Ebool_2ET ) )
      | ( V0f
        = ( ^ [V2b: $o] : c_2Ebool_2EF ) )
      | ( V0f
        = ( ^ [V3b: $o] : V3b ) )
      | ( V0f
        = ( ^ [V4b: $o] : ( c_2Ebool_2E_7E @ V4b ) ) ) ) ).

thf(thm_2Ebool_2EBOOL__FUN__INDUCT,axiom,
    ! [V0P: ( $o > $o ) > $o] :
      ( ( ( V0P
          @ ^ [V1b: $o] : c_2Ebool_2ET )
        & ( V0P
          @ ^ [V2b: $o] : c_2Ebool_2EF )
        & ( V0P
          @ ^ [V3b: $o] : V3b )
        & ( V0P
          @ ^ [V4b: $o] : ( c_2Ebool_2E_7E @ V4b ) ) )
     => ! [V5f: $o > $o] : ( V0P @ V5f ) ) ).

thf(thm_2Ebool_2Eliteral__case__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a] :
      ( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V0f @ V1x )
      = ( V0f @ V1x ) ) ).

thf(thm_2Ebool_2Eliteral__case__RAND,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27b > A_27c,V1N: A_27a > A_27b,V2M: A_27a] :
      ( ( V0P
        @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b
          @ ^ [V3x: A_27a] : ( V1N @ V3x )
          @ V2M ) )
      = ( c_2Ebool_2Eliteral__case @ A_27a @ A_27c
        @ ^ [V4x: A_27a] : ( V0P @ ( V1N @ V4x ) )
        @ V2M ) ) ).

thf(thm_2Ebool_2Eliteral__case__RATOR,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0b: A_27b,V1N: A_27a > A_27b > A_27c,V2M: A_27a] :
      ( ( c_2Ebool_2Eliteral__case @ A_27a @ ( A_27b > A_27c )
        @ ^ [V3x: A_27a] : ( V1N @ V3x )
        @ V2M
        @ V0b )
      = ( c_2Ebool_2Eliteral__case @ A_27a @ A_27c
        @ ^ [V4x: A_27a] : ( V1N @ V4x @ V0b )
        @ V2M ) ) ).

thf(thm_2Ebool_2Eliteral__case__CONG,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b,V2M: A_27a,V3N: A_27a] :
      ( ( ( V2M = V3N )
        & ! [V4x: A_27a] :
            ( ( V4x = V3N )
           => ( ( V0f @ V4x )
              = ( V1g @ V4x ) ) ) )
     => ( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V0f @ V2M )
        = ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V1g @ V3N ) ) ) ).

thf(thm_2Ebool_2Eliteral__case__id,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0u: A_27b,V1t: A_27b,V2a: A_27a] :
      ( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b
        @ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2a ) @ V1t @ V0u )
        @ V2a )
      = V1t ) ).

thf(thm_2Ebool_2EBOUNDED__THM,axiom,
    ! [V0v: $o] :
      ( ( c_2Ebool_2EBOUNDED @ V0v )
      = c_2Ebool_2ET ) ).

thf(thm_2Ebool_2ELCOMM__THM,axiom,
    ! [A_27a: $tType,V0f: A_27a > A_27a > A_27a] :
      ( ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
          ( ( V0f @ V1x @ ( V0f @ V2y @ V3z ) )
          = ( V0f @ ( V0f @ V1x @ V2y ) @ V3z ) )
     => ( ! [V4x: A_27a,V5y: A_27a] :
            ( ( V0f @ V4x @ V5y )
            = ( V0f @ V5y @ V4x ) )
       => ! [V6x: A_27a,V7y: A_27a,V8z: A_27a] :
            ( ( V0f @ V6x @ ( V0f @ V7y @ V8z ) )
            = ( V0f @ V7y @ ( V0f @ V6x @ V8z ) ) ) ) ) ).

thf(thm_2Ebool_2EDATATYPE__TAG__THM,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( c_2Ebool_2EDATATYPE @ A_27a @ V0x )
      = c_2Ebool_2ET ) ).

thf(thm_2Ebool_2EDATATYPE__BOOL,axiom,
    ! [V0bool: $o > $o > $o] :
      ( ( c_2Ebool_2EDATATYPE @ $o @ ( V0bool @ c_2Ebool_2ET @ c_2Ebool_2EF ) )
      = c_2Ebool_2ET ) ).

thf(thm_2Ebool_2EITSELF__UNIQUE,axiom,
    ! [A_27a: $tType,V0i: tyop_2Ebool_2Eitself @ A_27a] :
      ( V0i
      = ( c_2Ebool_2Ethe__value @ A_27a ) ) ).

thf(thm_2Ebool_2Eitself__Axiom,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0e: A_27b] :
    ? [V1f: ( tyop_2Ebool_2Eitself @ A_27a ) > A_27b] :
      ( ( V1f @ ( c_2Ebool_2Ethe__value @ A_27a ) )
      = V0e ) ).

thf(thm_2Ebool_2Eitself__induction,axiom,
    ! [A_27a: $tType,V0P: ( tyop_2Ebool_2Eitself @ A_27a ) > $o] :
      ( ( V0P @ ( c_2Ebool_2Ethe__value @ A_27a ) )
     => ! [V1i: tyop_2Ebool_2Eitself @ A_27a] : ( V0P @ V1i ) ) ).

thf(thm_2Ebool_2EFORALL__itself,axiom,
    ! [A_27a: $tType,V0P: ( tyop_2Ebool_2Eitself @ A_27a ) > $o] :
      ( ! [V1x: tyop_2Ebool_2Eitself @ A_27a] : ( V0P @ V1x )
    <=> ( V0P @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ).

thf(thm_2Ebool_2EEXISTS__itself,axiom,
    ! [A_27a: $tType,V0P: ( tyop_2Ebool_2Eitself @ A_27a ) > $o] :
      ( ? [V1x: tyop_2Ebool_2Eitself @ A_27a] : ( V0P @ V1x )
    <=> ( V0P @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ).

thf(thm_2Ebool_2EPULL__EXISTS,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ( ( ? [V2x: A_27a] : ( V0P @ V2x )
         => V1Q )
      <=> ! [V3x: A_27a] :
            ( ( V0P @ V3x )
           => V1Q ) )
      & ( ( ? [V4x: A_27a] : ( V0P @ V4x )
          & V1Q )
      <=> ? [V5x: A_27a] :
            ( ( V0P @ V5x )
            & V1Q ) )
      & ( ( V1Q
          & ? [V6x: A_27a] : ( V0P @ V6x ) )
      <=> ? [V7x: A_27a] :
            ( V1Q
            & ( V0P @ V7x ) ) ) ) ).

thf(thm_2Ebool_2EPULL__FORALL,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ( ( V1Q
         => ! [V2x: A_27a] : ( V0P @ V2x ) )
      <=> ! [V3x: A_27a] :
            ( V1Q
           => ( V0P @ V3x ) ) )
      & ( ( ! [V4x: A_27a] : ( V0P @ V4x )
          & V1Q )
      <=> ! [V5x: A_27a] :
            ( ( V0P @ V5x )
            & V1Q ) )
      & ( ( V1Q
          & ! [V6x: A_27a] : ( V0P @ V6x ) )
      <=> ! [V7x: A_27a] :
            ( V1Q
            & ( V0P @ V7x ) ) ) ) ).

thf(thm_2Ebool_2EPEIRCE,axiom,
    ! [V0Q: $o,V1P: $o] :
      ( ( ( V1P
         => V0Q )
       => V1P )
     => V1P ) ).

thf(thm_2Ebool_2EJRH__INDUCT__UTIL,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1t: A_27a] :
      ( ! [V2x: A_27a] :
          ( ( V2x = V1t )
         => ( V0P @ V2x ) )
     => ( c_2Ebool_2E_3F @ A_27a @ V0P ) ) ).

thf(thm_2Ebool_2EDISJ__EQ__IMP,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( V0A
        | V1B )
    <=> ( ( (~) @ V0A )
       => V1B ) ) ).

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