Theorem cbviinvw2

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

	Ref	Expression
Hypotheses	cbviinvw2.1	|- ( x = y -> C = D )
	cbviinvw2.2	|- ( x = y -> A = B )
Assertion	cbviinvw2	|- |^|_ x e. A C = |^|_ y e. B D

Proof

Step	Hyp	Ref	Expression
1		cbviinvw2.1	|- ( x = y -> C = D )
2		cbviinvw2.2	|- ( x = y -> A = B )
3	1	eleq2d	|- ( x = y -> ( t e. C <-> t e. D ) )
4	2 3	cbvralvw2	|- ( A. x e. A t e. C <-> A. y e. B t e. D )
5	4	abbii	|- { 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	3eqtr4i	|- |^|_ x e. A C = |^|_ y e. B D