% Mizar problem: t89_pboole,pboole,1196,18 
fof(t89_pboole,conjecture,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => r2_pboole(A,k4_pboole(A,B,C),k5_pboole(A,B,C)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[rc1_funct_1,commutativity_k2_pboole,idempotence_k2_pboole,commutativity_k5_pboole,reflexivity_r2_pboole,existence_m1_pboole,dt_k2_pboole,dt_k4_pboole,dt_k5_pboole,dt_m1_pboole,d12_pboole,t16_pboole]),
    [file(pboole,t89_pboole)]).

fof(rc1_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    []).

fof(commutativity_k2_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => k2_pboole(A,B,C) = k2_pboole(A,C,B) ) ),
    file(pboole,k2_pboole),
    []).

fof(idempotence_k2_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => k2_pboole(A,B,B) = B ) ),
    file(pboole,k2_pboole),
    []).

fof(commutativity_k5_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => k5_pboole(A,B,C) = k5_pboole(A,C,B) ) ),
    file(pboole,k5_pboole),
    []).

fof(reflexivity_r2_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => r2_pboole(A,B,B) ) ),
    file(pboole,r2_pboole),
    []).

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

fof(dt_k2_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => m1_pboole(k2_pboole(A,B,C),A) ) ),
    file(pboole,k2_pboole),
    []).

fof(dt_k4_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => m1_pboole(k4_pboole(A,B,C),A) ) ),
    file(pboole,k4_pboole),
    []).

fof(dt_k5_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => m1_pboole(k5_pboole(A,B,C),A) ) ),
    file(pboole,k5_pboole),
    []).

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

fof(d12_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => k5_pboole(A,B,C) = k2_pboole(A,k4_pboole(A,B,C),k4_pboole(A,C,B)) ) ) ),
    file(pboole,d12_pboole),
    []).

fof(t16_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ( r2_pboole(A,B,k2_pboole(A,B,C))
            & r2_pboole(A,C,k2_pboole(A,B,C)) ) ) ) ),
    file(pboole,t16_pboole),
    []).