Theorem iineq2i

Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003)

	Ref	Expression
Hypothesis	iuneq2i.1	$$⊢x\in A\to B=C$$
Assertion	iineq2i	$$⊢\bigcap _{x\in A}B=\bigcap _{x\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		iuneq2i.1	$$⊢x\in A\to B=C$$
2		iineq2	$$⊢\forall x\in AB=C\to \bigcap _{x\in A}B=\bigcap _{x\in A}C$$
3	2 1	mprg	$$⊢\bigcap _{x\in A}B=\bigcap _{x\in A}C$$