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Theorem soeq12d 37434
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 5052 . . 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 5053 . . 3 (𝐴 = 𝐵 → (𝑆 Or 𝐴𝑆 Or 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 Or 𝐴𝑆 Or 𝐵))
73, 6bitrd 268 1 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482   Or wor 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2916  df-in 3579  df-ss 3586  df-br 4652  df-po 5033  df-so 5034
This theorem is referenced by: (None)
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