Theorem eliind2

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	eliind2.1	$$⊢Ⅎx\phi$$
	eliind2.2	$$⊢\phi \to A\in V$$
	eliind2.3	$$⊢\left(\phi \wedge x\in B\right)\to A\in C$$
Assertion	eliind2	$$⊢\phi \to A\in \bigcap _{x\in B}C$$

Proof

Step	Hyp	Ref	Expression
1		eliind2.1	$$⊢Ⅎx\phi$$
2		eliind2.2	$$⊢\phi \to A\in V$$
3		eliind2.3	$$⊢\left(\phi \wedge x\in B\right)\to A\in C$$
4	3	ex	$$⊢\phi \to \left(x\in B\to A\in C\right)$$
5	1 4	ralrimi	$$⊢\phi \to \forall x\in BA\in C$$
6		eliin	$$⊢A\in V\to \left(A\in \bigcap _{x\in B}C\leftrightarrow \forall x\in BA\in C\right)$$
7	2 6	syl	$$⊢\phi \to \left(A\in \bigcap _{x\in B}C\leftrightarrow \forall x\in BA\in C\right)$$
8	5 7	mpbird	$$⊢\phi \to A\in \bigcap _{x\in B}C$$