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Theorem breqtrid 5111
Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
breqtrid.1 𝐴𝑅𝐵
breqtrid.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
breqtrid (𝜑𝐴𝑅𝐶)

Proof of Theorem breqtrid
StepHypRef Expression
1 breqtrid.1 . . 3 𝐴𝑅𝐵
21a1i 11 . 2 (𝜑𝐴𝑅𝐵)
3 breqtrid.2 . 2 (𝜑𝐵 = 𝐶)
42, 3breqtrd 5100 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   class class class wbr 5074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075
This theorem is referenced by:  breqtrrid  5112  phplem3OLD  9002  xlemul1a  13022  phicl2  16469  sinq12ge0  25665  siilem1  29213  nmbdfnlbi  30411  nmcfnlbi  30414  unierri  30466  leoprf2  30489  leoprf  30490  ballotlemic  32473  ballotlem1c  32474  sumnnodd  43171
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