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Theorem syl5breqr 4325
Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
syl5breqr.1  |-  A R B
syl5breqr.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
syl5breqr  |-  ( ph  ->  A R C )

Proof of Theorem syl5breqr
StepHypRef Expression
1 syl5breqr.1 . 2  |-  A R B
2 syl5breqr.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2446 . 2  |-  ( ph  ->  B  =  C )
41, 3syl5breq 4324 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364   class class class wbr 4289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290
This theorem is referenced by:  r1sdom  7977  alephordilem1  8239  mulge0  9853  xsubge0  11220  xmulgt0  11242  xmulge0  11243  xlemul1a  11247  sqlecan  11968  bernneq  11986  hashge1  12148  hashge2el2dif  12180  cnpart  12725  sqr0lem  12726  bitsfzo  13627  bitsmod  13628  bitsinv1lem  13633  pcge0  13924  prmreclem4  13976  prmreclem5  13977  isabvd  16885  abvtrivd  16905  nmolb2d  20197  nmoi  20207  nmoleub  20210  nmo0  20214  ovolge0  20864  itg1ge0a  21089  fta1g  21582  plyrem  21714  taylfval  21767  abelthlem2  21840  sinq12ge0  21913  relogrn  21956  logneg  21979  cxpge0  22071  amgmlem  22326  bposlem5  22570  lgsdir2lem2  22606  rpvmasumlem  22679  eupath2lem3  23519  eupath2  23520  blocnilem  24123  pjssge0ii  25004  unierri  25427  esumcst  26434  ballotlem5  26796  itgaddnclem2  28360  monotoddzzfi  29192  rmxypos  29199  rmygeid  29216  stoweidlem18  29722  stoweidlem55  29759  wallispi2lem1  29775  frgrawopreglem2  30547  pgrpgt2nabel  30669  isnzr2hash  30674
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