Theorem elabgt

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg .) (Contributed by NM, 7-Nov-2005) (Proof shortened by Andrew Salmon, 8-Jun-2011)

	Ref	Expression
Assertion	elabgt	|- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		nfcv	|- F/_ x A
2		nfab1	|- F/_ x { x | ph }
3	2	nfel2	|- F/ x A e. { x | ph }
4		nfv	|- F/ x ps
5	3 4	nfbi	|- F/ x ( A e. { x | ph } <-> ps )
6		pm5.5	|- ( x = A -> ( ( x = A -> ( A e. { x | ph } <-> ps ) ) <-> ( A e. { x | ph } <-> ps ) ) )
7	1 5 6	spcgf	|- ( A e. B -> ( A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) -> ( A e. { x | ph } <-> ps ) ) )
8		abid	|- ( x e. { x | ph } <-> ph )
9		eleq1	|- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
10	8 9	bitr3id	|- ( x = A -> ( ph <-> A e. { x | ph } ) )
11	10	bibi1d	|- ( x = A -> ( ( ph <-> ps ) <-> ( A e. { x | ph } <-> ps ) ) )
12	11	biimpd	|- ( x = A -> ( ( ph <-> ps ) -> ( A e. { x | ph } <-> ps ) ) )
13	12	a2i	|- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( A e. { x | ph } <-> ps ) ) )
14	13	alimi	|- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) )
15	7 14	impel	|- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )