let X1, X2 be set ; :: thesis: ( ( for x being object holds
( x in X1 iff ex y being object st [x,y] in X ) ) & ( for x being object holds
( x in X2 iff ex y being object st [x,y] in X ) ) implies X1 = X2 )
assume that
A3: for x being object holds
( x in X1 iff ex y being object st [x,y] in X ) and
A4: for x being object holds
( x in X2 iff ex y being object st [x,y] in X ) ; :: thesis: X1 = X2

now :: thesis: for x being object holds
( x in X1 iff x in X2 )

let x be object ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff ex y being object st [x,y] in X ) by A3;
hence ( x in X1 iff x in X2 ) by A4; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum