Theorem cbvdisjvw2

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

	Ref	Expression
Hypotheses	cbvdisjvw2.1	|- ( x = y -> C = D )
	cbvdisjvw2.2	|- ( x = y -> A = B )
Assertion	cbvdisjvw2	|- ( Disj_ x e. A C <-> Disj_ y e. B D )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjvw2.1	|- ( x = y -> C = D )
2		cbvdisjvw2.2	|- ( x = y -> A = B )
3	1	eleq2d	|- ( x = y -> ( t e. C <-> t e. D ) )
4	2 3	cbvrmovw2	|- ( E* x e. A t e. C <-> E* y e. B t e. D )
5	4	albii	|- ( 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	3bitr4i	|- ( Disj_ x e. A C <-> Disj_ y e. B D )