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		elab6g	|- ( A e. V -> ( A e. { x | ph } <-> A. x ( x = A -> ph ) ) )
3	1	pm5.74i	|- ( ( x = A -> ph ) <-> ( x = A -> ps ) )
4	3	albii	|- ( A. x ( x = A -> ph ) <-> A. x ( x = A -> ps ) )
5		19.23v	|- ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) )
6	4 5	bitri	|- ( A. x ( x = A -> ph ) <-> ( E. x x = A -> ps ) )
7		elisset	|- ( A e. V -> E. x x = A )
8		pm5.5	|- ( E. x x = A -> ( ( E. x x = A -> ps ) <-> ps ) )
9	7 8	syl	|- ( A e. V -> ( ( E. x x = A -> ps ) <-> ps ) )
10	6 9	syl5bb	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )
11	2 10	bitrd	|- ( A e. V -> ( A e. { x | ph } <-> ps ) )