%------------------------------------------------------------------------------
% File     : SEU709^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Conditionals
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! phi:o.! x:i.in x A -> (! y:i.in y A ->
%            in (if A phi x y) (setadjoin x (setadjoin y emptyset))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC211l [Bro08]

% Status   : Theorem
% Rating   : 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   15 (   5 unt;   9 typ;   5 def)
%            Number of atoms       :   30 (   9 equ;   0 cnn)
%            Maximal formula atoms :    8 (   5 avg)
%            Number of connectives :   61 (   1   ~;   0   |;   0   &;  44   @)
%                                         (   1 <=>;  15  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   4 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   9 usr;   6 con; 0-4 aty)
%            Number of variables   :   20 (   0   ^;  20   !;   0   ?;  20   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=265
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(emptyset_type,type,
    emptyset: $i ).

thf(setadjoin_type,type,
    setadjoin: $i > $i > $i ).

thf(setadjoinIL_type,type,
    setadjoinIL: $o ).

thf(setadjoinIL,definition,
    ( setadjoinIL
    = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).

thf(in__Cong_type,type,
    in__Cong: $o ).

thf(in__Cong,definition,
    ( in__Cong
    = ( ! [A: $i,B: $i] :
          ( ( A = B )
         => ! [Xx: $i,Xy: $i] :
              ( ( Xx = Xy )
             => ( ( in @ Xx @ A )
              <=> ( in @ Xy @ B ) ) ) ) ) ) ).

thf(secondinupair_type,type,
    secondinupair: $o ).

thf(secondinupair,definition,
    ( secondinupair
    = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).

thf(if_type,type,
    if: $i > $o > $i > $i > $i ).

thf(iftrue_type,type,
    iftrue: $o ).

thf(iftrue,definition,
    ( iftrue
    = ( ! [A: $i,Xphi: $o,Xx: $i] :
          ( ( in @ Xx @ A )
         => ! [Xy: $i] :
              ( ( in @ Xy @ A )
             => ( Xphi
               => ( ( if @ A @ Xphi @ Xx @ Xy )
                  = Xx ) ) ) ) ) ) ).

thf(iffalse_type,type,
    iffalse: $o ).

thf(iffalse,definition,
    ( iffalse
    = ( ! [A: $i,Xphi: $o,Xx: $i] :
          ( ( in @ Xx @ A )
         => ! [Xy: $i] :
              ( ( in @ Xy @ A )
             => ( ~ Xphi
               => ( ( if @ A @ Xphi @ Xx @ Xy )
                  = Xy ) ) ) ) ) ) ).

thf(iftrueorfalse,conjecture,
    ( setadjoinIL
   => ( in__Cong
     => ( secondinupair
       => ( iftrue
         => ( iffalse
           => ! [A: $i,Xphi: $o,Xx: $i] :
                ( ( in @ Xx @ A )
               => ! [Xy: $i] :
                    ( ( in @ Xy @ A )
                   => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------