Theorem elabgt 3347

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3351.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

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

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		nfcv 2764	. . 3 |- F/_ x A
2		nfab1 2766	. . . . 5 |- F/_ x { x | ph }
3	2	nfel2 2781	. . . 4 |- F/ x A e. { x | ph }
4		nfv 1843	. . . 4 |- F/ x ps
5	3, 4	nfbi 1833	. . 3 |- F/ x ( A e. { x | ph } <-> ps )
6		pm5.5 351	. . 3 |- ( x = A -> ( ( x = A -> ( A e. { x | ph } <-> ps ) ) <-> ( A e. { x | ph } <-> ps ) ) )
7	1, 5, 6	spcgf 3288	. 2 |- ( A e. B -> ( A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) -> ( A e. { x | ph } <-> ps ) ) )
8		abid 2610	. . . . . . 7 |- ( x e. { x | ph } <-> ph )
9		eleq1 2689	. . . . . . 7 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
10	8, 9	syl5bbr 274	. . . . . 6 |- ( x = A -> ( ph <-> A e. { x | ph } ) )
11	10	bibi1d 333	. . . . 5 |- ( x = A -> ( ( ph <-> ps ) <-> ( A e. { x | ph } <-> ps ) ) )
12	11	biimpd 219	. . . 4 |- ( x = A -> ( ( ph <-> ps ) -> ( A e. { x | ph } <-> ps ) ) )
13	12	a2i 14	. . 3 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( A e. { x | ph } <-> ps ) ) )
14	13	alimi 1739	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) )
15	7, 14	impel 485	1 |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  elrab3t  3362  dfrtrcl2  13802  abfmpeld  29454  abfmpel  29455  dftrcl3  38012  dfrtrcl3  38025