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 -> ( ph <-> ps ) )
Assertion	elabg	|- ( A e. V -> ( A e. { x | ph } <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		elabg.1	|- ( x = A -> ( ph <-> ps ) )
2	1	ax-gen	|- A. x ( x = A -> ( ph <-> ps ) )
3		elabgt	|- ( ( A e. V /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )
4	2 3	mpan2	|- ( A e. V -> ( A e. { x | ph } <-> ps ) )