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

Theorem eqbrtri 4021
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 4007 . 2 (𝐴𝑅𝐶𝐵𝑅𝐶)
41, 3mpbir 146 1 𝐴𝑅𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1353   class class class wbr 4000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001
This theorem is referenced by:  eqbrtrri  4023  3brtr4i  4030  exmidonfinlem  7186  neg1lt0  9013  halflt1  9122  3halfnz  9336  declei  9405  numlti  9406  faclbnd3  10704  geo2lim  11505  0.999...  11510  geoihalfsum  11511  fprodap0  11610  fprodap0f  11625  tan0  11720  cos2bnd  11749  sin4lt0  11755  eirraplem  11765  1nprm  12094  znnen  12379  tan4thpi  13922  zabsle1  14060  ex-fl  14126  trilpolemisumle  14435
  Copyright terms: Public domain W3C validator