Theorem rabeq12f

Description: Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018)

	Ref	Expression
Hypotheses	rabeq12f.1	|- F/_ x A
	rabeq12f.2	|- F/_ x B
Assertion	rabeq12f	|- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> { x e. A | ph } = { x e. B | ps } )

Proof

Step	Hyp	Ref	Expression
1		rabeq12f.1	|- F/_ x A
2		rabeq12f.2	|- F/_ x B
3		rabbi	|- ( A. x e. A ( ph <-> ps ) <-> { x e. A | ph } = { x e. A | ps } )
4	3	biimpi	|- ( A. x e. A ( ph <-> ps ) -> { x e. A | ph } = { x e. A | ps } )
5	1 2	rabeqf	|- ( A = B -> { x e. A | ps } = { x e. B | ps } )
6	4 5	sylan9eqr	|- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> { x e. A | ph } = { x e. B | ps } )