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Theorem eqbrtrdi 3967
Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
Hypotheses
Ref Expression
eqbrtrdi.1 (𝜑𝐴 = 𝐵)
eqbrtrdi.2 𝐵𝑅𝐶
Assertion
Ref Expression
eqbrtrdi (𝜑𝐴𝑅𝐶)

Proof of Theorem eqbrtrdi
StepHypRef Expression
1 eqbrtrdi.2 . 2 𝐵𝑅𝐶
2 eqbrtrdi.1 . . 3 (𝜑𝐴 = 𝐵)
32breq1d 3939 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mpbiri 167 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331   class class class wbr 3929
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 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  eqbrtrrdi  3968  pm54.43  7051  nn0ledivnn  9566  xltnegi  9630  leexp1a  10360  facwordi  10498  faclbnd3  10501  resqrexlemlo  10797  efap0  11395  dvds1  11562  en1top  12260  dvef  12871  trirec0  13298
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