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Theorem syl6eqbr 3830
Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
Hypotheses
Ref Expression
syl6eqbr.1 (𝜑𝐴 = 𝐵)
syl6eqbr.2 𝐵𝑅𝐶
Assertion
Ref Expression
syl6eqbr (𝜑𝐴𝑅𝐶)

Proof of Theorem syl6eqbr
StepHypRef Expression
1 syl6eqbr.2 . 2 𝐵𝑅𝐶
2 syl6eqbr.1 . . 3 (𝜑𝐴 = 𝐵)
32breq1d 3803 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mpbiri 166 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285   class class class wbr 3793
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 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794
This theorem is referenced by:  syl6eqbrr  3831  pm54.43  6518  nn0ledivnn  8919  xltnegi  8978  leexp1a  9628  facwordi  9764  faclbnd3  9767  resqrexlemlo  10037  dvds1  10398
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