ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl5eqbr GIF version

Theorem syl5eqbr 3844
Description: B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
syl5eqbr.1 𝐴 = 𝐵
syl5eqbr.2 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
syl5eqbr (𝜑𝐴𝑅𝐶)

Proof of Theorem syl5eqbr
StepHypRef Expression
1 syl5eqbr.2 . 2 (𝜑𝐵𝑅𝐶)
2 syl5eqbr.1 . 2 𝐴 = 𝐵
3 eqid 2083 . 2 𝐶 = 𝐶
41, 2, 33brtr4g 3843 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285   class class class wbr 3811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812
This theorem is referenced by:  xp1en  6467  caucvgprlemm  7128  intqfrac2  9613  m1modge3gt1  9665  bernneq2  9908  nno  10684  oddprmge3  10894  sqnprm  10895  oddennn  10983
  Copyright terms: Public domain W3C validator