Theorem iineq12dv

Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	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	iineq1d	|- ( ph -> |^|_ x e. A C = |^|_ x e. B C )
4	2	iineq2dv	|- ( ph -> |^|_ x e. B C = |^|_ x e. B D )
5	3 4	eqtrd	|- ( ph -> |^|_ x e. A C = |^|_ x e. B D )