Theorem cbviunvw2

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

	Ref	Expression
Hypotheses	cbviunvw2.1	|- ( x = y -> C = D )
	cbviunvw2.2	|- ( x = y -> A = B )
Assertion	cbviunvw2	|- U_ x e. A C = U_ y e. B D

Proof

Step	Hyp	Ref	Expression
1		cbviunvw2.1	|- ( x = y -> C = D )
2		cbviunvw2.2	|- ( x = y -> A = B )
3	1	eleq2d	|- ( x = y -> ( t e. C <-> t e. D ) )
4	2 3	cbvrexvw2	|- ( E. x e. A t e. C <-> E. y e. B t e. D )
5	4	abbii	|- { t | E. x e. A t e. C } = { t | E. y e. B t e. D }
6		df-iun	|- U_ x e. A C = { t | E. x e. A t e. C }
7		df-iun	|- U_ y e. B D = { t | E. y e. B t e. D }
8	5 6 7	3eqtr4i	|- U_ x e. A C = U_ y e. B D