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

Proof

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