Theorem iineq2d

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011)

	Ref	Expression
Hypotheses	iineq2d.1	$$⊢Ⅎx\phi$$
	iineq2d.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	iineq2d	$$⊢\phi \to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		iineq2d.1	$$⊢Ⅎx\phi$$
2		iineq2d.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
3	2	ex	$$⊢\phi \to \left(x\in A\to B=C\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in AB=C$$
5		iineq2	$$⊢\forall x\in AB=C\to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$
6	4 5	syl	$$⊢\phi \to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$