Theorem clel3g

Description: Alternate definition of membership in a set. (Contributed by NM, 13-Aug-2005)

	Ref	Expression
Assertion	clel3g	$$⊢B\in V\to \left(A\in B\leftrightarrow \exists x\left(x=B\wedge A\in x\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢x=B\to \left(A\in x\leftrightarrow A\in B\right)$$
2	1	ceqsexgv	$$⊢B\in V\to \left(\exists x\left(x=B\wedge A\in x\right)\leftrightarrow A\in B\right)$$
3	2	bicomd	$$⊢B\in V\to \left(A\in B\leftrightarrow \exists x\left(x=B\wedge A\in x\right)\right)$$