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Theorem ideqg 5024
 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 5002 . . . . 5
21brrelexi 4918 . . . 4
4 simpl 444 . . 3
53, 4jca 519 . 2
6 eleq1 2496 . . . . 5
76biimparc 474 . . . 4
8 elex 2964 . . . 4
97, 8syl 16 . . 3
10 simpl 444 . . 3
119, 10jca 519 . 2
12 eqeq1 2442 . . 3
13 eqeq2 2445 . . 3
14 df-id 4498 . . 3
1512, 13, 14brabg 4474 . 2
165, 11, 15pm5.21nd 869 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2956   class class class wbr 4212   cid 4493 This theorem is referenced by:  ideq  5025  ididg  5026  poleloe  5268  pltval  14417 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885
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