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) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 5-Oct-2024)

	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	2	elabg	$$⊢A\in \mathrm{V}\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$
4	1 3	ax-mp	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow \psi$$