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Theorem poeq2 5583
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 4034 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 poss 5581 . . 3 (𝐵𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
4 eqimss 4033 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 poss 5581 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
73, 6impbid 211 1 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wss 3941   Po wpo 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-in 3948  df-ss 3958  df-po 5579
This theorem is referenced by:  posn  5752  dfpo2  6286  frfi  9285  ipo0  43758
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