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Theorem eqbrtrdi 3962
Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
Hypotheses
Ref Expression
eqbrtrdi.1  |-  ( ph  ->  A  =  B )
eqbrtrdi.2  |-  B R C
Assertion
Ref Expression
eqbrtrdi  |-  ( ph  ->  A R C )

Proof of Theorem eqbrtrdi
StepHypRef Expression
1 eqbrtrdi.2 . 2  |-  B R C
2 eqbrtrdi.1 . . 3  |-  ( ph  ->  A  =  B )
32breq1d 3934 . 2  |-  ( ph  ->  ( A R C  <-> 
B R C ) )
41, 3mpbiri 167 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   class class class wbr 3924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925
This theorem is referenced by:  eqbrtrrdi  3963  pm54.43  7039  nn0ledivnn  9547  xltnegi  9611  leexp1a  10341  facwordi  10479  faclbnd3  10482  resqrexlemlo  10778  efap0  11372  dvds1  11540  en1top  12235  dvef  12845
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