Theorem cbviundavw2

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

	Ref	Expression
Hypotheses	cbviundavw2.1	|- ( ( ph /\ x = y ) -> C = D )
	cbviundavw2.2	|- ( ( ph /\ x = y ) -> A = B )
Assertion	cbviundavw2	|- ( ph -> U_ x e. A C = U_ y e. B D )

Proof

Step	Hyp	Ref	Expression
1		cbviundavw2.1	|- ( ( ph /\ x = y ) -> C = D )
2		cbviundavw2.2	|- ( ( ph /\ x = y ) -> A = B )
3	1	eleq2d	|- ( ( ph /\ x = y ) -> ( t e. C <-> t e. D ) )
4	3 2	cbvrexdva2	|- ( ph -> ( E. x e. A t e. C <-> E. y e. B t e. D ) )
5	4	abbidv	|- ( ph -> { 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	3eqtr4g	|- ( ph -> U_ x e. A C = U_ y e. B D )