Theorem cbviindavw2

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

	Ref	Expression
Hypotheses	cbviindavw2.1	|- ( ( ph /\ x = y ) -> C = D )
	cbviindavw2.2	|- ( ( ph /\ x = y ) -> A = B )
Assertion	cbviindavw2	|- ( ph -> |^|_ x e. A C = |^|_ y e. B D )

Proof

Step	Hyp	Ref	Expression
1		cbviindavw2.1	|- ( ( ph /\ x = y ) -> C = D )
2		cbviindavw2.2	|- ( ( ph /\ x = y ) -> A = B )
3	1	eleq2d	|- ( ( ph /\ x = y ) -> ( t e. C <-> t e. D ) )
4	3 2	cbvraldva2	|- ( ph -> ( A. x e. A t e. C <-> A. y e. B t e. D ) )
5	4	abbidv	|- ( ph -> { t | A. x e. A t e. C } = { t | A. y e. B t e. D } )
6		df-iin	|- |^|_ x e. A C = { t | A. x e. A t e. C }
7		df-iin	|- |^|_ y e. B D = { t | A. y e. B t e. D }
8	5 6 7	3eqtr4g	|- ( ph -> |^|_ x e. A C = |^|_ y e. B D )