Theorem elab3g

Description: Membership in a class abstraction, with a weaker antecedent than elabg . (Contributed by NM, 29-Aug-2006)

	Ref	Expression
Hypothesis	elab3g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	elab3g	$$⊢\left(\psi \to A\in B\right)\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elab3g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	elab3gf	$$⊢\left(\psi \to A\in B\right)\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$