Theorem rabbida3

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	rabbida3.1	|- F/ x ph
	rabbida3.2	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
Assertion	rabbida3	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Proof

Step	Hyp	Ref	Expression
1		rabbida3.1	|- F/ x ph
2		rabbida3.2	|- ( 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 } )