Theorem elab

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of Quine p. 44. (Contributed by NM, 1-Aug-1994)

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

Proof

Step	Hyp	Ref	Expression
1		elab.1	$$⊢A\in \mathrm{V}$$
2		elab.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		nfv	$$⊢Ⅎx\psi$$
4	3 1 2	elabf	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow \psi$$