ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqbrtri GIF version

Theorem eqbrtri 3949
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
eqbrtr.1 𝐴 = 𝐵
eqbrtr.2 𝐵𝑅𝐶
Assertion
Ref Expression
eqbrtri 𝐴𝑅𝐶

Proof of Theorem eqbrtri
StepHypRef Expression
1 eqbrtr.2 . 2 𝐵𝑅𝐶
2 eqbrtr.1 . . 3 𝐴 = 𝐵
32breq1i 3936 . 2 (𝐴𝑅𝐶𝐵𝑅𝐶)
41, 3mpbir 145 1 𝐴𝑅𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1331   class class class wbr 3929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  eqbrtrri  3951  3brtr4i  3958  exmidonfinlem  7054  neg1lt0  8840  halflt1  8949  3halfnz  9160  declei  9229  numlti  9230  faclbnd3  10501  geo2lim  11297  0.999...  11302  geoihalfsum  11303  tan0  11449  cos2bnd  11478  sin4lt0  11484  eirraplem  11494  1nprm  11806  znnen  11922  tan4thpi  12944  ex-fl  12996  trilpolemisumle  13292
  Copyright terms: Public domain W3C validator