Theorem eliin2

Description: Membership in indexed intersection. See eliincex for a counterexample showing that the precondition B =/= (/) cannot be simply dropped. eliin uses an alternative precondition (and it doesn't have a disjoint var constraint between B and x ; see eliin2f ). (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Assertion	eliin2	$$⊢B\ne \varnothing \to \left(A\in \bigcap _{x\in B}C\leftrightarrow \forall x\in BA\in C\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfcv	$$⊢\underset{\_}{Ⅎ}xB$$
2	1	eliin2f	$$⊢B\ne \varnothing \to \left(A\in \bigcap _{x\in B}C\leftrightarrow \forall x\in BA\in C\right)$$