Theorem iineq12dv

Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021) Remove DV conditions. (Revised by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	iineq12dv.1	|- ( ph -> A = B )
	iineq12dv.2	|- ( ( ph /\ x e. B ) -> C = D )
Assertion	iineq12dv	|- ( ph -> |^|_ x e. A C = |^|_ x e. B D )

Proof

Step	Hyp	Ref	Expression
1		iineq12dv.1	|- ( ph -> A = B )
2		iineq12dv.2	|- ( ( ph /\ x e. B ) -> C = D )
3	1	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
4	3	imbi1d	|- ( ph -> ( ( x e. A -> t e. C ) <-> ( x e. B -> t e. C ) ) )
5	4	ralbidv2	|- ( ph -> ( A. x e. A t e. C <-> A. x e. B t e. C ) )
6	5	abbidv	|- ( ph -> { t | A. x e. A t e. C } = { t | A. x e. B t e. C } )
7		df-iin	|- |^|_ x e. A C = { t | A. x e. A t e. C }
8		df-iin	|- |^|_ x e. B C = { t | A. x e. B t e. C }
9	6 7 8	3eqtr4g	|- ( ph -> |^|_ x e. A C = |^|_ x e. B C )
10	2	iineq2dv	|- ( ph -> |^|_ x e. B C = |^|_ x e. B D )
11	9 10	eqtrd	|- ( ph -> |^|_ x e. A C = |^|_ x e. B D )