Theorem elintrabg

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007)

	Ref	Expression
Assertion	elintrabg	$$⊢A\in V\to \left(A\in \bigcap \left\{x\in B|\phi \right\}\leftrightarrow \forall x\in B\left(\phi \to A\in x\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢y=A\to \left(y\in \bigcap \left\{x\in B|\phi \right\}\leftrightarrow A\in \bigcap \left\{x\in B|\phi \right\}\right)$$
2		eleq1	$$⊢y=A\to \left(y\in x\leftrightarrow A\in x\right)$$
3	2	imbi2d	$$⊢y=A\to \left(\left(\phi \to y\in x\right)\leftrightarrow \left(\phi \to A\in x\right)\right)$$
4	3	ralbidv	$$⊢y=A\to \left(\forall x\in B\left(\phi \to y\in x\right)\leftrightarrow \forall x\in B\left(\phi \to A\in x\right)\right)$$
5		vex	$$⊢y\in \mathrm{V}$$
6	5	elintrab	$$⊢y\in \bigcap \left\{x\in B|\phi \right\}\leftrightarrow \forall x\in B\left(\phi \to y\in x\right)$$
7	1 4 6	vtoclbg	$$⊢A\in V\to \left(A\in \bigcap \left\{x\in B|\phi \right\}\leftrightarrow \forall x\in B\left(\phi \to A\in x\right)\right)$$