Theorem iineq1d

Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypothesis	iineq1d.1	$$⊢\phi \to A=B$$
Assertion	iineq1d	$$⊢\phi \to \bigcap _{x\in A}C=\bigcap _{x\in B}C$$

Proof

Step	Hyp	Ref	Expression
1		iineq1d.1	$$⊢\phi \to A=B$$
2		iineq1	$$⊢A=B\to \bigcap _{x\in A}C=\bigcap _{x\in B}C$$
3	1 2	syl	$$⊢\phi \to \bigcap _{x\in A}C=\bigcap _{x\in B}C$$