Theorem eliind

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	eliind.a	$$⊢\phi \to A\in \bigcap _{x\in B}C$$
	eliind.k	$$⊢\phi \to K\in B$$
	eliind.d	$$⊢x=K\to \left(A\in C\leftrightarrow A\in D\right)$$
Assertion	eliind	$$⊢\phi \to A\in D$$

Proof

Step	Hyp	Ref	Expression
1		eliind.a	$$⊢\phi \to A\in \bigcap _{x\in B}C$$
2		eliind.k	$$⊢\phi \to K\in B$$
3		eliind.d	$$⊢x=K\to \left(A\in C\leftrightarrow A\in D\right)$$
4		eliin	$$⊢A\in \bigcap _{x\in B}C\to \left(A\in \bigcap _{x\in B}C\leftrightarrow \forall x\in BA\in C\right)$$
5	1 4	syl	$$⊢\phi \to \left(A\in \bigcap _{x\in B}C\leftrightarrow \forall x\in BA\in C\right)$$
6	1 5	mpbid	$$⊢\phi \to \forall x\in BA\in C$$
7	3 6 2	rspcdva	$$⊢\phi \to A\in D$$