Theorem cbvrabdavw

Description: Change bound variable in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025)

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

Proof

Step	Hyp	Ref	Expression
1		cbvrabdavw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3	2	adantl	|- ( ( ph /\ x = y ) -> ( x e. A <-> y e. A ) )
4	3 1	anbi12d	|- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. A /\ ch ) ) )
5	4	cbvabdavw	|- ( ph -> { x | ( x e. A /\ ps ) } = { y | ( y e. A /\ ch ) } )
6		df-rab	|- { x e. A | ps } = { x | ( x e. A /\ ps ) }
7		df-rab	|- { y e. A | ch } = { y | ( y e. A /\ ch ) }
8	5 6 7	3eqtr4g	|- ( ph -> { x e. A | ps } = { y e. A | ch } )