Theorem iineq2dv

Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004)

	Ref	Expression
Hypothesis	iuneq2dv.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	iineq2dv	$$⊢\phi \to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		iuneq2dv.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
2	1	ralrimiva	$$⊢\phi \to \forall x\in AB=C$$
3		iineq2	$$⊢\forall x\in AB=C\to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$
4	2 3	syl	$$⊢\phi \to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$