Theorem elint2

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999)

	Ref	Expression
Hypothesis	elint2.1	$$⊢A\in \mathrm{V}$$
Assertion	elint2	$$⊢A\in \bigcap B\leftrightarrow \forall x\in BA\in x$$

Proof

Step	Hyp	Ref	Expression
1		elint2.1	$$⊢A\in \mathrm{V}$$
2	1	elint	$$⊢A\in \bigcap B\leftrightarrow \forall x\left(x\in B\to A\in x\right)$$
3		df-ral	$$⊢\forall x\in BA\in x\leftrightarrow \forall x\left(x\in B\to A\in x\right)$$
4	2 3	bitr4i	$$⊢A\in \bigcap B\leftrightarrow \forall x\in BA\in x$$