Theorem elab2

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

	Ref	Expression
Hypotheses	elab2.1	$$⊢A\in \mathrm{V}$$
	elab2.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	elab2.3	$$⊢B=\left\{x|\phi \right\}$$
Assertion	elab2	$$⊢A\in B\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		elab2.1	$$⊢A\in \mathrm{V}$$
2		elab2.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		elab2.3	$$⊢B=\left\{x|\phi \right\}$$
4	2 3	elab2g	$$⊢A\in \mathrm{V}\to \left(A\in B\leftrightarrow \psi \right)$$
5	1 4	ax-mp	$$⊢A\in B\leftrightarrow \psi$$