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Theorem poeq2 5454
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 4003 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 poss 5452 . . 3 (𝐵𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
4 eqimss 4002 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 poss 5452 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
73, 6impbid 214 1 (𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wss 3913   Po wpo 5448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-ral 3130  df-v 3475  df-in 3920  df-ss 3930  df-po 5450
This theorem is referenced by:  posn  5613  frfi  8741  dfpo2  32999  ipo0  40936
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