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Theorem eqbrtrdi 4054
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 4025 . 2  |-  ( ph  ->  ( A R C  <-> 
B R C ) )
41, 3mpbiri 168 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363   class class class wbr 4015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016
This theorem is referenced by:  eqbrtrrdi  4055  pm54.43  7202  recapb  8641  nn0ledivnn  9780  xltnegi  9848  leexp1a  10588  facwordi  10733  faclbnd3  10736  resqrexlemlo  11035  efap0  11698  dvds1  11872  en1top  13817  dvef  14428  rpabscxpbnd  14599  zabsle1  14640  trirec0  15033
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