MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  poeq2 Unicode version

Theorem poeq2 4334
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 3244 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 poss 4332 . . 3  |-  ( B 
C_  A  ->  ( R  Po  A  ->  R  Po  B ) )
31, 2syl 15 . 2  |-  ( A  =  B  ->  ( R  Po  A  ->  R  Po  B ) )
4 eqimss 3243 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 poss 4332 . . 3  |-  ( A 
C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
64, 5syl 15 . 2  |-  ( A  =  B  ->  ( R  Po  B  ->  R  Po  A ) )
73, 6impbid 183 1  |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    C_ wss 3165    Po wpo 4328
This theorem is referenced by:  posn  4774  frfi  7118  dfpo2  24183  ipo0  27755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-in 3172  df-ss 3179  df-po 4330
  Copyright terms: Public domain W3C validator