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Theorem soeq12d 40871
Description: Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l (𝜑𝑅 = 𝑆)
weeq12d.r (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
soeq12d (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))

Proof of Theorem soeq12d
StepHypRef Expression
1 weeq12d.l . . 3 (𝜑𝑅 = 𝑆)
2 soeq1 5519 . . 3 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
31, 2syl 17 . 2 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐴))
4 weeq12d.r . . 3 (𝜑𝐴 = 𝐵)
5 soeq2 5520 . . 3 (𝐴 = 𝐵 → (𝑆 Or 𝐴𝑆 Or 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 Or 𝐴𝑆 Or 𝐵))
73, 6bitrd 278 1 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539   Or wor 5497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3431  df-in 3893  df-ss 3903  df-br 5074  df-po 5498  df-so 5499
This theorem is referenced by: (None)
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