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Theorem ideq 4836
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
Hypothesis
Ref Expression
ideq.1  |-  B  e. 
_V
Assertion
Ref Expression
ideq  |-  ( A  _I  B  <->  A  =  B )

Proof of Theorem ideq
StepHypRef Expression
1 ideq.1 . 2  |-  B  e. 
_V
2 ideqg 4835 . 2  |-  ( B  e.  _V  ->  ( A  _I  B  <->  A  =  B ) )
31, 2ax-mp 8 1  |-  ( A  _I  B  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    _I cid 4304
This theorem is referenced by:  dmi  4893  resieq  4965  resiexg  4997  iss  4998  imai  5027  issref  5056  intasym  5058  asymref  5059  intirr  5061  poirr2  5067  cnvi  5085  coi1  5188  dffv2  5592  idssen  6906  dflt2  10482  opsrtoslem2  16226  hausdiag  17339  hauseqlcld  17340  ex-id  20821  relexpindlem  24036  dfso2  24111  dfpo2  24112  idsset  24430  dfon3  24432  elfix  24443  dffix2  24445  dffun10  24453  elfuns  24454  brsingle  24456  brapply  24477  brsuccf  24480  dfrdg4  24488  restidsing  25076  ipo0  27652  ifr0  27653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696
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