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Theorem syl5eqbrr 3839
Description: B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
Hypotheses
Ref Expression
syl5eqbrr.1  |-  B  =  A
syl5eqbrr.2  |-  ( ph  ->  B R C )
Assertion
Ref Expression
syl5eqbrr  |-  ( ph  ->  A R C )

Proof of Theorem syl5eqbrr
StepHypRef Expression
1 syl5eqbrr.2 . 2  |-  ( ph  ->  B R C )
2 syl5eqbrr.1 . 2  |-  B  =  A
3 eqid 2083 . 2  |-  C  =  C
41, 2, 33brtr3g 3836 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   class class class wbr 3805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806
This theorem is referenced by:  enpr1g  6366  recexprlem1ssl  6937  addgt0  7671  addgegt0  7672  addgtge0  7673  addge0  7674  expge1  9662  ncoprmgcdne1b  10678  phicl2  10797
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