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Theorem breqtrid 5189
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 5178 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   class class class wbr 5152
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153
This theorem is referenced by:  breqtrrid  5190  phplem3OLD  9252  xlemul1a  13309  phicl2  16746  sinq12ge0  26471  siilem1  30689  nmbdfnlbi  31887  nmcfnlbi  31890  unierri  31942  leoprf2  31965  leoprf  31966  ballotlemic  34167  ballotlem1c  34168  sumnnodd  45065
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