MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl5breqr Structured version   Unicode version

Theorem syl5breqr 4328
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 2448 . 2  |-  ( ph  ->  B  =  C )
41, 3syl5breq 4327 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   class class class wbr 4292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293
This theorem is referenced by:  r1sdom  7981  alephordilem1  8243  mulge0  9857  xsubge0  11224  xmulgt0  11246  xmulge0  11247  xlemul1a  11251  sqlecan  11972  bernneq  11990  hashge1  12152  hashge2el2dif  12184  cnpart  12729  sqr0lem  12730  bitsfzo  13631  bitsmod  13632  bitsinv1lem  13637  pcge0  13928  prmreclem4  13980  prmreclem5  13981  isabvd  16905  abvtrivd  16925  nmolb2d  20297  nmoi  20307  nmoleub  20310  nmo0  20314  ovolge0  20964  itg1ge0a  21189  fta1g  21639  plyrem  21771  taylfval  21824  abelthlem2  21897  sinq12ge0  21970  relogrn  22013  logneg  22036  cxpge0  22128  amgmlem  22383  bposlem5  22627  lgsdir2lem2  22663  rpvmasumlem  22736  eupath2lem3  23600  eupath2  23601  blocnilem  24204  pjssge0ii  25085  unierri  25508  esumcst  26514  ballotlem5  26882  itgaddnclem2  28451  monotoddzzfi  29283  rmxypos  29290  rmygeid  29307  stoweidlem18  29813  stoweidlem55  29850  wallispi2lem1  29866  frgrawopreglem2  30638  pgrpgt2nabel  30769  isnzr2hash  30774
  Copyright terms: Public domain W3C validator