Theorem raleqbidv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007) Remove usage of ax-10 , ax-11 , and ax-12 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023)

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

Proof

Step	Hyp	Ref	Expression
1		raleqbidv.1	|- ( ph -> A = B )
2		raleqbidv.2	|- ( ph -> ( ps <-> ch ) )
3	1	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
4	3 2	imbi12d	|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )
5	4	ralbidv2	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )