% Mizar problem: t2_recdef_2,recdef_2,79,57 
fof(t2_recdef_2,conjecture,(
    ! [A,B,C,D] : 
      ( r2_hidden(A,k3_zfmisc_1(B,C,D))
     => ( r2_hidden(k1_recdef_2(A),B)
        & r2_hidden(k2_recdef_2(A),C)
        & r2_hidden(k3_recdef_2(A),D) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,d1_recdef_2,d2_recdef_2,d3_recdef_2,dt_k1_recdef_2,dt_k1_xboole_0,dt_k2_recdef_2,dt_k3_mcart_1,dt_k3_recdef_2,dt_k3_zfmisc_1,dt_m1_subset_1,existence_m1_subset_1,fc1_xboole_0,fc5_subset_1,rc1_xboole_0,rc2_xboole_0,t15_domain_1,t1_subset,t2_subset,t35_mcart_1,t6_boole,t7_boole,t8_boole]),
    [file(recdef_2,t2_recdef_2)]).

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

fof(d1_recdef_2,axiom,(
    ! [A] : 
      ( ? [B,C,D] : A = k3_mcart_1(B,C,D)
     => ! [B] : 
          ( B = k1_recdef_2(A)
        <=> ! [C,D,E] : 
              ( A = k3_mcart_1(C,D,E)
             => B = C ) ) ) ),
    file(recdef_2,d1_recdef_2),
    []).

fof(d2_recdef_2,axiom,(
    ! [A] : 
      ( ? [B,C,D] : A = k3_mcart_1(B,C,D)
     => ! [B] : 
          ( B = k2_recdef_2(A)
        <=> ! [C,D,E] : 
              ( A = k3_mcart_1(C,D,E)
             => B = D ) ) ) ),
    file(recdef_2,d2_recdef_2),
    []).

fof(d3_recdef_2,axiom,(
    ! [A] : 
      ( ? [B,C,D] : A = k3_mcart_1(B,C,D)
     => ! [B] : 
          ( B = k3_recdef_2(A)
        <=> ! [C,D,E] : 
              ( A = k3_mcart_1(C,D,E)
             => B = E ) ) ) ),
    file(recdef_2,d3_recdef_2),
    []).

fof(dt_k1_recdef_2,axiom,(
    $true ),
    file(recdef_2,k1_recdef_2),
    []).

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

fof(dt_k2_recdef_2,axiom,(
    $true ),
    file(recdef_2,k2_recdef_2),
    []).

fof(dt_k3_mcart_1,axiom,(
    $true ),
    file(mcart_1,k3_mcart_1),
    []).

fof(dt_k3_recdef_2,axiom,(
    $true ),
    file(recdef_2,k3_recdef_2),
    []).

fof(dt_k3_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k3_zfmisc_1),
    []).

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

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(fc5_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C) )
     => ~ v1_xboole_0(k3_zfmisc_1(A,B,C)) ) ),
    file(subset_1,fc5_subset_1),
    []).

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(t15_domain_1,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( ~ v1_xboole_0(C)
         => ! [D] : 
              ( ~ v1_xboole_0(D)
             => ( r2_hidden(A,k3_zfmisc_1(B,C,D))
              <=> ? [E] : 
                    ( m1_subset_1(E,B)
                    & ? [F] : 
                        ( m1_subset_1(F,C)
                        & ? [G] : 
                            ( m1_subset_1(G,D)
                            & A = k3_mcart_1(E,F,G) ) ) ) ) ) ) ) ),
    file(domain_1,t15_domain_1),
    []).

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(t35_mcart_1,axiom,(
    ! [A,B,C] : 
      ( ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0 )
    <=> k3_zfmisc_1(A,B,C) != k1_xboole_0 ) ),
    file(mcart_1,t35_mcart_1),
    []).

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),
    []).