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Theorem syl6eqbr 3843
Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
Hypotheses
Ref Expression
syl6eqbr.1  |-  ( ph  ->  A  =  B )
syl6eqbr.2  |-  B R C
Assertion
Ref Expression
syl6eqbr  |-  ( ph  ->  A R C )

Proof of Theorem syl6eqbr
StepHypRef Expression
1 syl6eqbr.2 . 2  |-  B R C
2 syl6eqbr.1 . . 3  |-  ( ph  ->  A  =  B )
32breq1d 3816 . 2  |-  ( ph  ->  ( A R C  <-> 
B R C ) )
41, 3mpbiri 166 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   class class class wbr 3806
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 2987  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807
This theorem is referenced by:  syl6eqbrr  3844  pm54.43  6554  nn0ledivnn  8955  xltnegi  9014  leexp1a  9664  facwordi  9800  faclbnd3  9803  resqrexlemlo  10084  dvds1  10445
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