let A1, A2 be set ; :: thesis: ( ( for x being set holds
( x in A1 iff ( x in X & x in Y ) ) ) & ( for x being set holds
( x in A2 iff ( x in X & x in Y ) ) ) implies A1 = A2 )
assume that
A5: for x being set holds
( x in A1 iff ( x in X & x in Y ) ) and
A6: for x being set holds
( x in A2 iff ( x in X & x in Y ) ) ; :: thesis: A1 = A2

now

let x be set ; :: thesis: ( x in A1 iff x in A2 )
( x in A1 iff ( x in X & x in Y ) ) by A5;
hence ( x in A1 iff x in A2 ) by A6; :: thesis: verum

end;

hence A1 = A2 by TARSKI:1; :: thesis: verum