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Theorem poeq2 4222
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 3152 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 poss 4220 . . 3 (𝐵𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐵))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
4 eqimss 3151 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 poss 4220 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
73, 6impbid 128 1 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wss 3071   Po wpo 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-in 3077  df-ss 3084  df-po 4218
This theorem is referenced by: (None)
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