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Theorem ideq 4852
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 4851 . 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 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    _I cid 4320
This theorem is referenced by:  dmi  4909  resieq  4981  resiexg  5013  iss  5014  imai  5043  issref  5072  intasym  5074  asymref  5075  intirr  5077  poirr2  5083  cnvi  5101  coi1  5204  dffv2  5608  idssen  6922  dflt2  10498  opsrtoslem2  16242  hausdiag  17355  hauseqlcld  17356  ex-id  20837  relexpindlem  24051  dfso2  24182  dfpo2  24183  idsset  24501  dfon3  24503  elfix  24514  dffix2  24516  dffun10  24524  elfuns  24525  brsingle  24527  brapply  24548  brsuccf  24551  dfrdg4  24560  restidsing  25179  ipo0  27755  ifr0  27756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712
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