Theorem iineq1i

Description: Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	iineq1i.1	|- A = B
Assertion	iineq1i	|- |^|_ x e. A C = |^|_ x e. B C

Proof

Step	Hyp	Ref	Expression
1		iineq1i.1	|- A = B
2	1	eleq2i	|- ( x e. A <-> x e. B )
3	2	imbi1i	|- ( ( x e. A -> t e. C ) <-> ( x e. B -> t e. C ) )
4	3	ralbii2	|- ( A. x e. A t e. C <-> A. x e. B t e. C )
5	4	abbii	|- { t | A. x e. A t e. C } = { t | A. x e. B t e. C }
6		df-iin	|- |^|_ x e. A C = { t | A. x e. A t e. C }
7		df-iin	|- |^|_ x e. B C = { t | A. x e. B t e. C }
8	5 6 7	3eqtr4i	|- |^|_ x e. A C = |^|_ x e. B C