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

Theorem syl5breqr 4251
Description: B 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 2443 . 2  |-  ( ph  ->  B  =  C )
41, 3syl5breq 4250 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   class class class wbr 4215
This theorem is referenced by:  r1sdom  7703  alephordilem1  7959  mulge0  9550  xsubge0  10845  xmulgt0  10867  xmulge0  10868  xlemul1a  10872  sqlecan  11492  bernneq  11510  hashge1  11668  cnpart  12050  sqr0lem  12051  bitsfzo  12952  bitsmod  12953  bitsinv1lem  12958  pcge0  13240  prmreclem4  13292  prmreclem5  13293  isabvd  15913  abvtrivd  15933  nmolb2d  18757  nmoi  18767  nmoleub  18770  nmo0  18774  ovolge0  19382  itg1ge0a  19606  fta1g  20095  plyrem  20227  taylfval  20280  abelthlem2  20353  sinq12ge0  20421  relogrn  20464  logneg  20487  cxpge0  20579  amgmlem  20833  bposlem5  21077  lgsdir2lem2  21113  rpvmasumlem  21186  eupath2lem3  21706  eupath2  21707  blocnilem  22310  pjssge0ii  23189  unierri  23612  esumcst  24460  ballotlem5  24762  itgaddnclem2  26278  monotoddzzfi  27019  rmxypos  27026  rmygeid  27043  stoweidlem18  27757  stoweidlem55  27794  wallispi2lem1  27810  frgrawopreglem2  28508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216
  Copyright terms: Public domain W3C validator