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Theorem poeq2 4028
 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 2992 . . 3 (A = BBA)
2 poss 4026 . . 3 (BA → (𝑅 Po A𝑅 Po B))
31, 2syl 14 . 2 (A = B → (𝑅 Po A𝑅 Po B))
4 eqimss 2991 . . 3 (A = BAB)
5 poss 4026 . . 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 1242   ⊆ wss 2911   Po wpo 4022 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925  df-po 4024 This theorem is referenced by: (None)
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