% Mizar problem: t2_xboole_0,xboole_0,183,68 
fof(t2_xboole_0,conjecture,(
    ! [A,B,C] : 
      ( ! [D] : 
          ( ~ r2_hidden(D,A)
        <=> ( r2_hidden(D,B)
          <=> r2_hidden(D,C) ) )
     => A = k5_xboole_0(B,C) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k2_xboole_0,commutativity_k5_xboole_0,d2_xboole_0,d4_xboole_0,d6_xboole_0,dt_k2_xboole_0,dt_k4_xboole_0,dt_k5_xboole_0,idempotence_k2_xboole_0,t2_tarski]),
    [file(xboole_0,t2_xboole_0)]).

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

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(commutativity_k5_xboole_0,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k5_xboole_0(B,A) ),
    file(xboole_0,k5_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(d4_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    []).

fof(d6_xboole_0,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) ),
    file(xboole_0,d6_xboole_0),
    []).

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

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    []).

fof(dt_k5_xboole_0,axiom,(
    $true ),
    file(xboole_0,k5_xboole_0),
    []).

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

fof(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).