Theorem cbvdisjdavw2

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

	Ref	Expression
Hypotheses	cbvdisjdavw2.1	|- ( ( ph /\ x = y ) -> C = D )
	cbvdisjdavw2.2	|- ( ( ph /\ x = y ) -> A = B )
Assertion	cbvdisjdavw2	|- ( ph -> ( Disj_ x e. A C <-> Disj_ y e. B D ) )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjdavw2.1	|- ( ( ph /\ x = y ) -> C = D )
2		cbvdisjdavw2.2	|- ( ( ph /\ x = y ) -> A = B )
3	1	eleq2d	|- ( ( ph /\ x = y ) -> ( t e. C <-> t e. D ) )
4	3 2	cbvrmodavw2	|- ( ph -> ( E* x e. A t e. C <-> E* y e. B t e. D ) )
5	4	albidv	|- ( ph -> ( A. t E* x e. A t e. C <-> A. t E* y e. B t e. D ) )
6		df-disj	|- ( Disj_ x e. A C <-> A. t E* x e. A t e. C )
7		df-disj	|- ( Disj_ y e. B D <-> A. t E* y e. B t e. D )
8	5 6 7	3bitr4g	|- ( ph -> ( Disj_ x e. A C <-> Disj_ y e. B D ) )