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Theorem breqtrid 5067
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 5056 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542   class class class wbr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-un 3848  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031
This theorem is referenced by:  breqtrrid  5068  phplem3  8748  xlemul1a  12764  phicl2  16205  sinq12ge0  25253  siilem1  28786  nmbdfnlbi  29984  nmcfnlbi  29987  unierri  30039  leoprf2  30062  leoprf  30063  ballotlemic  32043  ballotlem1c  32044  sumnnodd  42713
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