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Theorem poeq2 4297
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 3210 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 poss 4295 . . 3 (𝐵𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐵))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
4 eqimss 3209 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 poss 4295 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
73, 6impbid 129 1 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wss 3129   Po wpo 4291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3135  df-ss 3142  df-po 4293
This theorem is referenced by: (None)
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