Theorem elabgt 3243

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3247.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
elabgt

Distinct variable groups:   ,   ,

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		abid 2444	. . . . . . 7
2		eleq1 2529	. . . . . . 7
3	1, 2	syl5bbr 259	. . . . . 6
4	3	bibi1d 319	. . . . 5
5	4	biimpd 207	. . . 4
6	5	a2i 13	. . 3
7	6	alimi 1633	. 2
8		nfcv 2619	. . . 4
9		nfab1 2621	. . . . . 6
10	9	nfel2 2637	. . . . 5
11		nfv 1707	. . . . 5
12	10, 11	nfbi 1934	. . . 4
13		pm5.5 336	. . . 4
14	8, 12, 13	spcgf 3189	. . 3
15	14	imp 429	. 2
16	7, 15	sylan2 474	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  {cab 2442

This theorem is referenced by:  elrab3t  3256  abfmpeld  27492  abfmpel  27493  dfrtrcl2  29071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111