Theorem rabeqbida

Description: Version of rabeqbidva with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbida.nf	|- F/ x ph
2		rabeqbida.1	|- ( ph -> A = B )
3		rabeqbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
4	1 3	rabbida	|- ( ph -> { x e. A | ps } = { x e. A | ch } )
5	1 2	rabeqd	|- ( ph -> { x e. A | ch } = { x e. B | ch } )
6	4 5	eqtrd	|- ( ph -> { x e. A | ps } = { x e. B | ch } )