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Theorem poeq2 5004
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 3642 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 poss 5002 . . 3 (𝐵𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
4 eqimss 3641 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 poss 5002 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
73, 6impbid 202 1 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wss 3560   Po wpo 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-ral 2917  df-in 3567  df-ss 3574  df-po 5000
This theorem is referenced by:  posn  5153  frfi  8150  dfpo2  31344  ipo0  38102
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