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

Theorem eqbrtrri 4606
 Description: Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
Hypotheses
Ref Expression
eqbrtrr.1 𝐴 = 𝐵
eqbrtrr.2 𝐴𝑅𝐶
Assertion
Ref Expression
eqbrtrri 𝐵𝑅𝐶

Proof of Theorem eqbrtrri
StepHypRef Expression
1 eqbrtrr.1 . . 3 𝐴 = 𝐵
21eqcomi 2619 . 2 𝐵 = 𝐴
3 eqbrtrr.2 . 2 𝐴𝑅𝐶
42, 3eqbrtri 4604 1 𝐵𝑅𝐶
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584 This theorem is referenced by:  3brtr3i  4612  expnass  12832  faclbnd4lem1  12942  sqrt2gt1lt2  13863  cos1bnd  14756  cos2bnd  14757  2strstr1  15812  prdsvalstr  15936  ovolre  23100  pige3  24073  atan1  24455  log2ublem1  24473  sqrtlim  24499  bposlem8  24816  chebbnd1  24961  konigsberg  26514  norm-ii-i  27378  nmopadji  28333  unierri  28347  ballotlem2  29877  pigt3  32572  stoweidlem26  38919  wallispilem5  38962
 Copyright terms: Public domain W3C validator