Theorem elab2g

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995)

	Ref	Expression
Hypotheses	elab2g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	elab2g.2	$$⊢B=\left\{x|\phi \right\}$$
Assertion	elab2g	$$⊢A\in V\to \left(A\in B\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elab2g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		elab2g.2	$$⊢B=\left\{x|\phi \right\}$$
3	2	eleq2i	$$⊢A\in B\leftrightarrow A\in \left\{x|\phi \right\}$$
4	1	elabg	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$
5	3 4	syl5bb	$$⊢A\in V\to \left(A\in B\leftrightarrow \psi \right)$$