Theorem rabbida2

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

	Ref	Expression
Hypotheses	rabbida2.1	|- F/ x ph
	rabbida2.2	|- ( ph -> A = B )
	rabbida2.3	|- ( ph -> ( ps <-> ch ) )
Assertion	rabbida2	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Proof

Step	Hyp	Ref	Expression
1		rabbida2.1	|- F/ x ph
2		rabbida2.2	|- ( ph -> A = B )
3		rabbida2.3	|- ( ph -> ( ps <-> ch ) )
4	2	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
5	4 3	anbi12d	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
6	1 5	abbid	|- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
7		df-rab	|- { x e. A | ps } = { x | ( x e. A /\ ps ) }
8		df-rab	|- { x e. B | ch } = { x | ( x e. B /\ ch ) }
9	6 7 8	3eqtr4g	|- ( ph -> { x e. A | ps } = { x e. B | ch } )