Theorem ralbi12f

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

	Ref	Expression
Hypotheses	ralbi12f.1	|- F/_ x A
	ralbi12f.2	|- F/_ x B
Assertion	ralbi12f	|- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> ( A. x e. A ph <-> A. x e. B ps ) )

Proof

Step	Hyp	Ref	Expression
1		ralbi12f.1	|- F/_ x A
2		ralbi12f.2	|- F/_ x B
3		ralbi	|- ( A. x e. A ( ph <-> ps ) -> ( A. x e. A ph <-> A. x e. A ps ) )
4	1 2	raleqf	|- ( A = B -> ( A. x e. A ps <-> A. x e. B ps ) )
5	3 4	sylan9bbr	|- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> ( A. x e. A ph <-> A. x e. B ps ) )