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Theorem breqtrrdi 3965
Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
breqtrrdi.1 (𝜑𝐴𝑅𝐵)
breqtrrdi.2 𝐶 = 𝐵
Assertion
Ref Expression
breqtrrdi (𝜑𝐴𝑅𝐶)

Proof of Theorem breqtrrdi
StepHypRef Expression
1 breqtrrdi.1 . 2 (𝜑𝐴𝑅𝐵)
2 breqtrrdi.2 . . 3 𝐶 = 𝐵
32eqcomi 2141 . 2 𝐵 = 𝐶
41, 3breqtrdi 3964 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331   class class class wbr 3924
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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925
This theorem is referenced by:  enpr2d  6704  fiunsnnn  6768  unsnfi  6800  eninl  6975  eninr  6976  difinfinf  6979  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  dju1en  7062  djucomen  7065  djuassen  7066  xpdjuen  7067  gtndiv  9139  intqfrac2  10085  uzenom  10191  xrmaxiflemval  11012  ege2le3  11366  eirraplem  11472  2strstr1g  12051  lmcn2  12438  dveflem  12844  tangtx  12908  sbthom  13210
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