let A1, A2 be set ; :: thesis: ( ( for x being object holds
( x in A1 iff ( x in X & x in Y ) ) ) & ( for x being object holds
( x in A2 iff ( x in X & x in Y ) ) ) implies A1 = A2 )
assume that
A3: for x being object holds
( x in A1 iff ( x in X & x in Y ) ) and
A4: for x being object holds
( x in A2 iff ( x in X & x in Y ) ) ; :: thesis: A1 = A2

now :: thesis: for x being object holds
( x in A1 iff x in A2 )

let x be object ; :: thesis: ( x in A1 iff x in A2 )
( x in A1 iff ( x in X & x in Y ) ) by A3;
hence ( x in A1 iff x in A2 ) by A4; :: thesis: verum

end;

hence A1 = A2 by TARSKI:2; :: thesis: verum