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Theorem poeq2 4011
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poeq2 (A = B → (𝑅 Po A𝑅 Po B))

Proof of Theorem poeq2
StepHypRef Expression
1 eqimss2 2975 . . 3 (A = BBA)
2 poss 4009 . . 3 (BA → (𝑅 Po A𝑅 Po B))
31, 2syl 14 . 2 (A = B → (𝑅 Po A𝑅 Po B))
4 eqimss 2974 . . 3 (A = BAB)
5 poss 4009 . . 3 (AB → (𝑅 Po B𝑅 Po A))
64, 5syl 14 . 2 (A = B → (𝑅 Po B𝑅 Po A))
73, 6impbid 120 1 (A = B → (𝑅 Po A𝑅 Po B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228  wss 2894   Po wpo 4005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-in 2901  df-ss 2908  df-po 4007
This theorem is referenced by: (None)
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