% Mizar problem: t21_mboolean,mboolean,553,41 
fof(t21_mboolean,conjecture,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ( r1_pboole(A,B,k2_mboolean(A,C))
          <=> ? [D] : 
                ( m1_pboole(D,A)
                & r1_pboole(A,B,D)
                & r1_pboole(A,D,C) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,d2_mboolean,d4_pboole,d4_tarski,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_mboolean,dt_k3_tarski,dt_m1_pboole,dt_m1_subset_1,existence_m1_pboole,existence_m1_subset_1,fc1_xboole_0,rc1_xboole_0,rc2_xboole_0,s3_pboole__e3_31_1__mboolean,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole]),
    [file(mboolean,t21_mboolean)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(d2_mboolean,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ( C = k2_mboolean(A,B)
          <=> ! [D] : 
                ( r2_hidden(D,A)
               => k1_funct_1(C,D) = k3_tarski(k1_funct_1(B,D)) ) ) ) ) ),
    file(mboolean,d2_mboolean),
    []).

fof(d4_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ( r1_pboole(A,B,C)
          <=> ! [D] : 
                ( r2_hidden(D,A)
               => r2_hidden(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) ),
    file(pboole,d4_pboole),
    []).

fof(d4_tarski,axiom,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    []).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k2_mboolean,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => m1_pboole(k2_mboolean(A,B),A) ) ),
    file(mboolean,k2_mboolean),
    []).

fof(dt_k3_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    []).

fof(dt_m1_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B) ) ) ),
    file(pboole,m1_pboole),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_pboole,axiom,(
    ! [A] : 
    ? [B] : m1_pboole(B,A) ),
    file(pboole,m1_pboole),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(s3_pboole__e3_31_1__mboolean,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ( ! [D] : ~ ( r2_hidden(D,A)
            & ! [E] : ~ ( r2_hidden(k1_funct_1(B,D),E)
                & r2_hidden(E,k1_funct_1(C,D)) ) )
       => ? [D] : 
            ( m1_pboole(D,A)
            & ! [E] : 
                ( r2_hidden(E,A)
               => ( r2_hidden(k1_funct_1(B,E),k1_funct_1(D,E))
                  & r2_hidden(k1_funct_1(D,E),k1_funct_1(C,E)) ) ) ) ) ) ),
    file(mboolean,s3_pboole__e3_31_1__mboolean),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).