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Theorem poeq2 4745
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 3509 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 poss 4743 . . 3  |-  ( B 
C_  A  ->  ( R  Po  A  ->  R  Po  B ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( R  Po  A  ->  R  Po  B ) )
4 eqimss 3508 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 poss 4743 . . 3  |-  ( A 
C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
64, 5syl 16 . 2  |-  ( A  =  B  ->  ( R  Po  B  ->  R  Po  A ) )
73, 6impbid 191 1  |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    C_ wss 3428    Po wpo 4739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-ral 2800  df-in 3435  df-ss 3442  df-po 4741
This theorem is referenced by:  posn  5007  frfi  7660  dfpo2  27701  ipo0  29845
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