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 e. _V
	elab.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	elab	|- ( A e. { x | ph } <-> ps )

Proof

Step	Hyp	Ref	Expression
1		elab.1	|- A e. _V
2		elab.2	|- ( x = A -> ( ph <-> ps ) )
3	2	elabg	|- ( A e. _V -> ( A e. { x | ph } <-> ps ) )
4	1 3	ax-mp	|- ( A e. { x | ph } <-> ps )