Theorem ideqg 5159

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ideqg

Proof of Theorem ideqg

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reli 5135	. . . . 5
2	1	brrelexi 5045	. . . 4
3	2	adantl 466	. . 3
4		simpl 457	. . 3
5	3, 4	jca 532	. 2
6		eleq1 2529	. . . . 5
7	6	biimparc 487	. . . 4
8		elex 3118	. . . 4
9	7, 8	syl 16	. . 3
10		simpl 457	. . 3
11	9, 10	jca 532	. 2
12		eqeq1 2461	. . 3
13		eqeq2 2472	. . 3
14		df-id 4800	. . 3
15	12, 13, 14	brabg 4771	. 2
16	5, 11, 15	pm5.21nd 900	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109   class class class wbr 4452  cid 4795

This theorem is referenced by:  ideq  5160  ididg  5161  restidsing  5335  poleloe  5406  pltval  15590  tglngne  23937  tgelrnln  24010  opeldifid  27456  fourierdlem42  31931

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011