Theorem rabbida4

Description: Version of rabbidva2 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

	Ref	Expression
Hypotheses	rabbida4.nf	|- F/ x ph
	rabbida4.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
Assertion	rabbida4	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Proof

Step	Hyp	Ref	Expression
1		rabbida4.nf	|- F/ x ph
2		rabbida4.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
3	1 2	abbid	|- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
4		df-rab	|- { x e. A | ps } = { x | ( x e. A /\ ps ) }
5		df-rab	|- { x e. B | ch } = { x | ( x e. B /\ ch ) }
6	3 4 5	3eqtr4g	|- ( ph -> { x e. A | ps } = { x e. B | ch } )