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Theorem ideq 5196
 Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
Hypothesis
Ref Expression
ideq.1 𝐵 ∈ V
Assertion
Ref Expression
ideq (𝐴 I 𝐵𝐴 = 𝐵)

Proof of Theorem ideq
StepHypRef Expression
1 ideq.1 . 2 𝐵 ∈ V
2 ideqg 5195 . 2 (𝐵 ∈ V → (𝐴 I 𝐵𝐴 = 𝐵))
31, 2ax-mp 5 1 (𝐴 I 𝐵𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   I cid 4948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045 This theorem is referenced by:  dmi  5261  resieq  5327  iss  5367  restidsing  5377  restidsingOLD  5378  imai  5397  issref  5428  intasym  5430  asymref  5431  intirr  5433  poirr2  5439  cnvi  5456  xpdifid  5481  coi1  5568  dffv2  6181  isof1oidb  6474  resiexg  6994  idssen  7886  dflt2  11857  relexpindlem  13651  opsrtoslem2  19306  hausdiag  21258  hauseqlcld  21259  metustid  22169  ltgov  25292  ex-id  26683  dfso2  30897  dfpo2  30898  idsset  31167  dfon3  31169  elfix  31180  dffix2  31182  sscoid  31190  dffun10  31191  elfuns  31192  brsingle  31194  brapply  31215  brsuccf  31218  dfrdg4  31228  undmrnresiss  36929  dffrege99  37276  ipo0  37674  ifr0  37675  fourierdlem42  39042
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