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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.
StepHypRef Expression
1 reli 5135 . . . . 5
21brrelexi 5045 . . . 4
4 simpl 457 . . 3
53, 4jca 532 . 2
6 eleq1 2529 . . . . 5
76biimparc 487 . . . 4
8 elex 3118 . . . 4
97, 8syl 16 . . 3
10 simpl 457 . . 3
119, 10jca 532 . 2
12 eqeq1 2461 . . 3
13 eqeq2 2472 . . 3
14 df-id 4800 . . 3
1512, 13, 14brabg 4771 . 2
165, 11, 15pm5.21nd 900 1
 Colors of variables: wff setvar class 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
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