% Mizar problem: t120_zfmisc_1,zfmisc_1,1405,77 
fof(t120_zfmisc_1,conjecture,(
    ! [A,B,C] : 
      ( k2_zfmisc_1(k2_xboole_0(A,B),C) = k2_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
      & k2_zfmisc_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k2_tarski,commutativity_k2_xboole_0,d2_xboole_0,d5_tarski,dt_k1_tarski,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k4_tarski,fc1_zfmisc_1,fc2_xboole_0,fc3_xboole_0,idempotence_k2_xboole_0,l55_zfmisc_1,rc1_xboole_0,rc2_xboole_0,t102_zfmisc_1,t107_zfmisc_1,t112_zfmisc_1]),
    [file(zfmisc_1,t120_zfmisc_1)]).

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

fof(commutativity_k2_tarski,axiom,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    []).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(d2_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    []).

fof(d5_tarski,axiom,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    []).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    []).

fof(dt_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    []).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    []).

fof(fc1_zfmisc_1,axiom,(
    ! [A,B] : ~ v1_xboole_0(k4_tarski(A,B)) ),
    file(zfmisc_1,fc1_zfmisc_1),
    []).

fof(fc2_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    []).

fof(fc3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(l55_zfmisc_1,axiom,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,l55_zfmisc_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(t102_zfmisc_1,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,k2_zfmisc_1(B,C))
      & ! [D,E] : k4_tarski(D,E) != A ) ),
    file(zfmisc_1,t102_zfmisc_1),
    []).

fof(t107_zfmisc_1,axiom,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
     => r2_hidden(k4_tarski(B,A),k2_zfmisc_1(D,C)) ) ),
    file(zfmisc_1,t107_zfmisc_1),
    []).

fof(t112_zfmisc_1,axiom,(
    ! [A,B] : 
      ( ( ! [C] : ~ ( r2_hidden(C,A)
            & ! [D,E] : C != k4_tarski(D,E) )
        & ! [C] : ~ ( r2_hidden(C,B)
            & ! [D,E] : C != k4_tarski(D,E) )
        & ! [C,D] : 
            ( r2_hidden(k4_tarski(C,D),A)
          <=> r2_hidden(k4_tarski(C,D),B) ) )
     => A = B ) ),
    file(zfmisc_1,t112_zfmisc_1),
    []).