Theorem cbvsbcdavw2

Description: Change bound variable of a class substitution. General version of cbvsbcdavw . Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvsbcdavw2.1	|- ( ph -> A = B )
	cbvsbcdavw2.2	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion	cbvsbcdavw2	|- ( ph -> ( [. A / x ]. ps <-> [. B / y ]. ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbvsbcdavw2.1	|- ( ph -> A = B )
2		cbvsbcdavw2.2	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
3	2	cbvabdavw	|- ( ph -> { x | ps } = { y | ch } )
4	1 3	eleq12d	|- ( ph -> ( A e. { x | ps } <-> B e. { y | ch } ) )
5		df-sbc	|- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc	|- ( [. B / y ]. ch <-> B e. { y | ch } )
7	4 5 6	3bitr4g	|- ( ph -> ( [. A / x ]. ps <-> [. B / y ]. ch ) )