Theorem iineq12i

Description: Equality theorem for indexed intersection. Inference version. General version of iineq1i . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	iineq12i.1	|- A = B
	iineq12i.2	|- C = D
Assertion	iineq12i	|- |^|_ x e. A C = |^|_ x e. B D

Proof

Step	Hyp	Ref	Expression
1		iineq12i.1	|- A = B
2		iineq12i.2	|- C = D
3	2	eleq2i	|- ( t e. C <-> t e. D )
4	1 3	raleqbii	|- ( A. x e. A t e. C <-> A. x e. B t e. D )
5	4	abbii	|- { t | A. x e. A t e. C } = { t | A. x e. B t e. D }
6		df-iin	|- |^|_ x e. A C = { t | A. x e. A t e. C }
7		df-iin	|- |^|_ x e. B D = { t | A. x e. B t e. D }
8	5 6 7	3eqtr4i	|- |^|_ x e. A C = |^|_ x e. B D