Theorem cbvdisjdavw

Description: Change bound variable in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvdisjdavw.1	|- ( ( ph /\ x = y ) -> B = C )
Assertion	cbvdisjdavw	|- ( ph -> ( Disj_ x e. A B <-> Disj_ y e. A C ) )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjdavw.1	|- ( ( ph /\ x = y ) -> B = C )
2	1	eleq2d	|- ( ( ph /\ x = y ) -> ( t e. B <-> t e. C ) )
3	2	cbvrmodavw	|- ( ph -> ( E* x e. A t e. B <-> E* y e. A t e. C ) )
4	3	albidv	|- ( ph -> ( A. t E* x e. A t e. B <-> A. t E* y e. A t e. C ) )
5		df-disj	|- ( Disj_ x e. A B <-> A. t E* x e. A t e. B )
6		df-disj	|- ( Disj_ y e. A C <-> A. t E* y e. A t e. C )
7	4 5 6	3bitr4g	|- ( ph -> ( Disj_ x e. A B <-> Disj_ y e. A C ) )