Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl6breq GIF version

Theorem syl6breq 3830
 Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
syl6breq.1 (𝜑𝐴𝑅𝐵)
syl6breq.2 𝐵 = 𝐶
Assertion
Ref Expression
syl6breq (𝜑𝐴𝑅𝐶)

Proof of Theorem syl6breq
StepHypRef Expression
1 syl6breq.1 . 2 (𝜑𝐴𝑅𝐵)
2 eqid 2056 . 2 𝐴 = 𝐴
3 syl6breq.2 . 2 𝐵 = 𝐶
41, 2, 33brtr3g 3822 1 (𝜑𝐴𝑅𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   class class class wbr 3791 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792 This theorem is referenced by:  syl6breqr  3831
 Copyright terms: Public domain W3C validator