ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poeq2 Unicode version

Theorem poeq2 4083
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poeq2  |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )

Proof of Theorem poeq2
StepHypRef Expression
1 eqimss2 3061 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 poss 4081 . . 3  |-  ( B 
C_  A  ->  ( R  Po  A  ->  R  Po  B ) )
31, 2syl 14 . 2  |-  ( A  =  B  ->  ( R  Po  A  ->  R  Po  B ) )
4 eqimss 3060 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 poss 4081 . . 3  |-  ( A 
C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
64, 5syl 14 . 2  |-  ( A  =  B  ->  ( R  Po  B  ->  R  Po  A ) )
73, 6impbid 127 1  |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    C_ wss 2982    Po wpo 4077
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  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-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-in 2988  df-ss 2995  df-po 4079
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator