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Theorem ideq 5006
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 5005 . 2  |-  ( B  e.  _V  ->  ( A  _I  B  <->  A  =  B ) )
31, 2ax-mp 5 1  |-  ( A  _I  B  <->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1872   _Vcvv 3080   class class class wbr 4423    _I cid 4763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860
This theorem is referenced by:  dmi  5068  resieq  5134  iss  5171  restidsing  5180  imai  5199  issref  5232  intasym  5234  asymref  5235  intirr  5237  poirr2  5243  cnvi  5259  xpdifid  5284  coi1  5370  dffv2  5955  resiexg  6744  idssen  7625  dflt2  11455  relexpindlem  13127  opsrtoslem2  18708  hausdiag  20659  hauseqlcld  20660  metustid  21568  ltgov  24641  ex-id  25883  dfso2  30402  dfpo2  30403  idsset  30663  dfon3  30665  elfix  30676  dffix2  30678  sscoid  30686  dffun10  30687  elfuns  30688  brsingle  30690  brapply  30711  brsuccf  30714  dfrdg4  30724  undmrnresiss  36181  dffrege99  36529  ipo0  36773  ifr0  36774  fourierdlem42  37953  fourierdlem42OLD  37954
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