Theorem raleqbidva

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		raleqbidva.1	|- ( ph -> A = B )
2		raleqbidva.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	ralbidva	|- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
4	1	raleqdv	|- ( ph -> ( A. x e. A ch <-> A. x e. B ch ) )
5	3 4	bitrd	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )