Theorem rabeqbidva

Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017) Remove DV conditions. (Revised by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbidva.1	|- ( ph -> A = B )
2		rabeqbidva.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	rabbidva	|- ( ph -> { x e. A | ps } = { x e. A | ch } )
4	1	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
5	4	anbi1d	|- ( ph -> ( ( x e. A /\ ch ) <-> ( x e. B /\ ch ) ) )
6	5	rabbidva2	|- ( ph -> { x e. A | ch } = { x e. B | ch } )
7	3 6	eqtrd	|- ( ph -> { x e. A | ps } = { x e. B | ch } )