Theorem cbvabdavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvabdavw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2	1	cbvsbdavw	|- ( ph -> ( [ t / x ] ps <-> [ t / y ] ch ) )
3		df-clab	|- ( t e. { x | ps } <-> [ t / x ] ps )
4		df-clab	|- ( t e. { y | ch } <-> [ t / y ] ch )
5	2 3 4	3bitr4g	|- ( ph -> ( t e. { x | ps } <-> t e. { y | ch } ) )
6	5	eqrdv	|- ( ph -> { x | ps } = { y | ch } )