Theorem rabbidva2

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017)

	Ref	Expression
Hypothesis	rabbidva2.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
Assertion	rabbidva2	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Proof

Step	Hyp	Ref	Expression
1		rabbidva2.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
2	1	abbidv	|- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
3		df-rab	|- { x e. A | ps } = { x | ( x e. A /\ ps ) }
4		df-rab	|- { x e. B | ch } = { x | ( x e. B /\ ch ) }
5	2 3 4	3eqtr4g	|- ( ph -> { x e. A | ps } = { x e. B | ch } )