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Theorem ideqg 4515
 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 4493 . . . . 5
21brrelexi 4410 . . . 4
4 simpl 107 . . 3
53, 4jca 300 . 2
6 eleq1 2142 . . . . 5
76biimparc 293 . . . 4
8 elex 2611 . . . 4
97, 8syl 14 . . 3
10 simpl 107 . . 3
119, 10jca 300 . 2
12 eqeq1 2088 . . 3
13 eqeq2 2091 . . 3
14 df-id 4056 . . 3
1512, 13, 14brabg 4032 . 2
165, 11, 15pm5.21nd 859 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wceq 1285   wcel 1434  cvv 2602   class class class wbr 3793   cid 4051 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378 This theorem is referenced by:  ideq  4516  ididg  4517  poleloe  4754
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