Theorem elabg

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of Quine p. 44. (Contributed by NM, 14-Apr-1995) Avoid ax-13 . (Revised by SN, 23-Nov-2022) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 5-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		elabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	ax-gen	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)$$
3		elabgt	$$⊢\left(A\in V\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$
4	2 3	mpan2	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \psi \right)$$