Theorem elab4g

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012)

	Ref	Expression
Hypotheses	elab4g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	elab4g.2	$$⊢B=\left\{x|\phi \right\}$$
Assertion	elab4g	$$⊢A\in B\leftrightarrow \left(A\in \mathrm{V}\wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elab4g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		elab4g.2	$$⊢B=\left\{x|\phi \right\}$$
3		elex	$$⊢A\in B\to A\in \mathrm{V}$$
4	1 2	elab2g	$$⊢A\in \mathrm{V}\to \left(A\in B\leftrightarrow \psi \right)$$
5	3 4	biadanii	$$⊢A\in B\leftrightarrow \left(A\in \mathrm{V}\wedge \psi \right)$$