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Theorem syl5eqbr 3822
Description: B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
syl5eqbr.1 𝐴 = 𝐵
syl5eqbr.2 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
syl5eqbr (𝜑𝐴𝑅𝐶)

Proof of Theorem syl5eqbr
StepHypRef Expression
1 syl5eqbr.2 . 2 (𝜑𝐵𝑅𝐶)
2 syl5eqbr.1 . 2 𝐴 = 𝐵
3 eqid 2054 . 2 𝐶 = 𝐶
41, 2, 33brtr4g 3821 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1257   class class class wbr 3789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790
This theorem is referenced by:  xp1en  6325  caucvgprlemm  6794  intqfrac2  9234  m1modge3gt1  9286  bernneq2  9502  nno  10181
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