Theorem elintd

Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021)

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

Proof

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