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Theorem breqtrid 5178
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 5167 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   class class class wbr 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142
This theorem is referenced by:  breqtrrid  5179  phplem3OLD  9221  xlemul1a  13273  phicl2  16710  sinq12ge0  26398  siilem1  30613  nmbdfnlbi  31811  nmcfnlbi  31814  unierri  31866  leoprf2  31889  leoprf  31890  ballotlemic  34035  ballotlem1c  34036  sumnnodd  44918
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