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

Theorem breqtrrid 3966
Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
breqtrrid.1  |-  A R B
breqtrrid.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
breqtrrid  |-  ( ph  ->  A R C )

Proof of Theorem breqtrrid
StepHypRef Expression
1 breqtrrid.1 . 2  |-  A R B
2 breqtrrid.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2145 . 2  |-  ( ph  ->  B  =  C )
41, 3breqtrid 3965 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  xsubge0  9671  xposdif  9672  bernneq  10419  trilpolemclim  13282  trilpolemlt1  13287
  Copyright terms: Public domain W3C validator