% Mizar ND problem: t2_boole,boole,43,16 
fof(dh_c1_2__boole,definition,
    ( k3_xboole_0(c1_2__boole,k1_xboole_0) = k1_xboole_0
   => ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ),
    introduced(definition,[new_symbol(c1_2__boole),file(boole,c1_2__boole)]),
    [interesting(0.8),axiom,file(boole,c1_2__boole)]).

fof(rc1_xboole_0,theorem,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]).

fof(rc2_xboole_0,theorem,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]).

fof(commutativity_k3_xboole_0,theorem,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]).

fof(idempotence_k3_xboole_0,theorem,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]).

fof(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(dt_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]).

fof(dt_c1_2__boole,assumption,(
    $true ),
    introduced(assumption,[file(boole,c1_2__boole)]),
    [interesting(0.8),axiom,file(boole,c1_2__boole)]).

fof(fc1_xboole_0,theorem,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]).

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(dt_c1_2_1__boole,assumption,(
    $true ),
    introduced(assumption,[file(boole,c1_2_1__boole)]),
    [interesting(0.65),axiom,file(boole,c1_2_1__boole)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.9),axiom,file(tarski,d3_tarski)]).

fof(dh_c1_2_1__boole,definition,
    ( ~ ( r2_hidden(c1_2_1__boole,k3_xboole_0(c1_2__boole,k1_xboole_0))
        & ~ r2_hidden(c1_2_1__boole,k1_xboole_0) )
   => ! [A] : 
        ~ ( r2_hidden(A,k3_xboole_0(c1_2__boole,k1_xboole_0))
          & ~ r2_hidden(A,k1_xboole_0) ) ),
    introduced(definition,[new_symbol(c1_2_1__boole),file(boole,c1_2_1__boole)]),
    [interesting(0.65),axiom,file(boole,c1_2_1__boole)]).

fof(e1_2_1__boole,assumption,(
    r2_hidden(c1_2_1__boole,k3_xboole_0(c1_2__boole,k1_xboole_0)) ),
    introduced(assumption,[file(boole,e1_2_1__boole)]),
    [interesting(0.65),axiom,file(boole,e1_2_1__boole)]).

fof(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d3_xboole_0)]).

fof(e2_2_1__boole,plain,(
    r2_hidden(c1_2_1__boole,k1_xboole_0) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__boole,dt_c1_2_1__boole,e1_2_1__boole])],[rc1_xboole_0,rc2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k3_xboole_0,dt_c1_2__boole,dt_c1_2_1__boole,fc1_xboole_0,e1_2_1__boole,d3_xboole_0]),
    [interesting(0.65),file(boole,e2_2_1__boole),[file(boole,e2_2_1__boole)]]).

fof(i3_2_1__boole,theorem,(
    $true ),
    introduced(tautology,[file(boole,i3_2_1__boole)]),
    [interesting(0.65),trivial,file(boole,i3_2_1__boole)]).

fof(i2_2_1__boole,plain,(
    r2_hidden(c1_2_1__boole,k1_xboole_0) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__boole,dt_c1_2_1__boole,e1_2_1__boole])],[e2_2_1__boole,i3_2_1__boole]),
    [interesting(0.65),file(boole,i2_2_1__boole),[file(boole,i2_2_1__boole)]]).

fof(i1_2_1__boole,plain,(
    ~ ( r2_hidden(c1_2_1__boole,k3_xboole_0(c1_2__boole,k1_xboole_0))
      & ~ r2_hidden(c1_2_1__boole,k1_xboole_0) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__boole,dt_c1_2_1__boole]),discharge_asm(discharge,[e1_2_1__boole])],[e1_2_1__boole,i2_2_1__boole]),
    [interesting(0.65),file(boole,i1_2_1__boole),[file(boole,i1_2_1__boole)]]).

fof(i1_2_1_tmp__boole,plain,(
    ~ ( r2_hidden(c1_2_1__boole,k3_xboole_0(c1_2__boole,k1_xboole_0))
      & ~ r2_hidden(c1_2_1__boole,k1_xboole_0) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__boole]),discharge_asm(discharge,[dt_c1_2_1__boole])],[dt_c1_2_1__boole,i1_2_1__boole]),
    [interesting(0.8),e1_2__boole]).

fof(e1_2__boole,plain,(
    r1_tarski(k3_xboole_0(c1_2__boole,k1_xboole_0),k1_xboole_0) ),
    inference(let,[status(thm),assumptions([dt_c1_2__boole])],[i1_2_1_tmp__boole,rc1_xboole_0,rc2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k3_xboole_0,dt_c1_2__boole,fc1_xboole_0,d3_tarski,dh_c1_2_1__boole]),
    [interesting(0.8),file(boole,e1_2__boole),[file(boole,e1_2__boole)]]).

fof(dt_c2_2__boole,assumption,(
    $true ),
    introduced(assumption,[file(boole,c2_2__boole)]),
    [interesting(0.8),axiom,file(boole,c2_2__boole)]).

fof(dh_c2_2__boole,definition,
    ( ~ ( r2_hidden(c2_2__boole,k1_xboole_0)
        & ~ r2_hidden(c2_2__boole,k3_xboole_0(c1_2__boole,k1_xboole_0)) )
   => ! [A] : 
        ~ ( r2_hidden(A,k1_xboole_0)
          & ~ r2_hidden(A,k3_xboole_0(c1_2__boole,k1_xboole_0)) ) ),
    introduced(definition,[new_symbol(c2_2__boole),file(boole,c2_2__boole)]),
    [interesting(0.8),axiom,file(boole,c2_2__boole)]).

fof(e2_2__boole,assumption,(
    r2_hidden(c2_2__boole,k1_xboole_0) ),
    introduced(assumption,[file(boole,e2_2__boole)]),
    [interesting(0.8),axiom,file(boole,e2_2__boole)]).

fof(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d1_xboole_0)]).

fof(e3_2__boole,plain,(
    r2_hidden(c2_2__boole,k3_xboole_0(c1_2__boole,k1_xboole_0)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__boole,dt_c2_2__boole,e2_2__boole])],[rc1_xboole_0,rc2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k3_xboole_0,dt_c1_2__boole,dt_c2_2__boole,fc1_xboole_0,e2_2__boole,d1_xboole_0]),
    [interesting(0.8),file(boole,e3_2__boole),[file(boole,e3_2__boole)]]).

fof(i5_2__boole,theorem,(
    $true ),
    introduced(tautology,[file(boole,i5_2__boole)]),
    [interesting(0.8),trivial,file(boole,i5_2__boole)]).

fof(i4_2__boole,plain,(
    r2_hidden(c2_2__boole,k3_xboole_0(c1_2__boole,k1_xboole_0)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__boole,dt_c2_2__boole,e2_2__boole])],[e3_2__boole,i5_2__boole]),
    [interesting(0.8),file(boole,i4_2__boole),[file(boole,i4_2__boole)]]).

fof(i3_2__boole,plain,(
    ~ ( r2_hidden(c2_2__boole,k1_xboole_0)
      & ~ r2_hidden(c2_2__boole,k3_xboole_0(c1_2__boole,k1_xboole_0)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__boole,dt_c2_2__boole]),discharge_asm(discharge,[e2_2__boole])],[e2_2__boole,i4_2__boole]),
    [interesting(0.8),file(boole,i3_2__boole),[file(boole,i3_2__boole)]]).

fof(i3_2_tmp__boole,plain,(
    ~ ( r2_hidden(c2_2__boole,k1_xboole_0)
      & ~ r2_hidden(c2_2__boole,k3_xboole_0(c1_2__boole,k1_xboole_0)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__boole]),discharge_asm(discharge,[dt_c2_2__boole])],[dt_c2_2__boole,i3_2__boole]),
    [interesting(0.8),i2_2__boole]).

fof(i2_2__boole,plain,(
    r1_tarski(k1_xboole_0,k3_xboole_0(c1_2__boole,k1_xboole_0)) ),
    inference(let,[status(thm),assumptions([dt_c1_2__boole])],[i3_2_tmp__boole,rc1_xboole_0,rc2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k3_xboole_0,dt_c1_2__boole,fc1_xboole_0,d3_tarski,dh_c2_2__boole]),
    [interesting(0.8),file(boole,i2_2__boole),[file(boole,i2_2__boole)]]).

fof(i1_2__boole,plain,(
    k3_xboole_0(c1_2__boole,k1_xboole_0) = k1_xboole_0 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__boole])],[rc1_xboole_0,rc2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k3_xboole_0,dt_c1_2__boole,fc1_xboole_0,d10_xboole_0,e1_2__boole,i2_2__boole]),
    [interesting(0.8),file(boole,i1_2__boole),[file(boole,i1_2__boole)]]).

fof(i1_2_tmp__boole,plain,(
    k3_xboole_0(c1_2__boole,k1_xboole_0) = k1_xboole_0 ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__boole])],[dt_c1_2__boole,i1_2__boole]),
    [interesting(1),t2_boole]).

fof(t2_boole,theorem,(
    ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ),
    inference(let,[status(thm),assumptions([])],[i1_2_tmp__boole,dh_c1_2__boole]),
    [interesting(1),file(boole,t2_boole),[file(boole,t2_boole)]]).