MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  breqtrid Structured version   Visualization version   GIF version

Theorem breqtrid 5143
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 5132 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542   class class class wbr 5106
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107
This theorem is referenced by:  breqtrrid  5144  phplem3OLD  9166  xlemul1a  13213  phicl2  16645  sinq12ge0  25881  siilem1  29835  nmbdfnlbi  31033  nmcfnlbi  31036  unierri  31088  leoprf2  31111  leoprf  31112  ballotlemic  33163  ballotlem1c  33164  sumnnodd  43957
  Copyright terms: Public domain W3C validator