% Mizar problem: t107_xboole_1,xboole_1,1272,50 
fof(t107_xboole_1,conjecture,(
    ! [A,B,C] : 
      ( r1_tarski(A,k5_xboole_0(B,C))
    <=> ( r1_tarski(A,k2_xboole_0(B,C))
        & r1_xboole_0(A,k3_xboole_0(B,C)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k2_xboole_0,commutativity_k3_xboole_0,commutativity_k5_xboole_0,d6_xboole_0,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_xboole_0,dt_k5_xboole_0,idempotence_k2_xboole_0,idempotence_k3_xboole_0,l98_xboole_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,t106_xboole_1,t86_xboole_1]),
    [file(xboole_1,t107_xboole_1)]).

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_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_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(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_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_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(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    []).

fof(l98_xboole_1,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) ),
    file(xboole_1,l98_xboole_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(symmetry_r1_xboole_0,axiom,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) ),
    file(xboole_0,r1_xboole_0),
    []).

fof(t106_xboole_1,axiom,(
    ! [A,B,C] : 
      ( r1_tarski(A,k4_xboole_0(B,C))
     => ( r1_tarski(A,B)
        & r1_xboole_0(A,C) ) ) ),
    file(xboole_1,t106_xboole_1),
    []).

fof(t86_xboole_1,axiom,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_xboole_0(A,C) )
     => r1_tarski(A,k4_xboole_0(B,C)) ) ),
    file(xboole_1,t86_xboole_1),
    []).