Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  soeq12d Structured version   Visualization version   GIF version

Theorem soeq12d 38110
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 5206 . . 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 5207 . . 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 1632   Or wor 5186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-in 3722  df-ss 3729  df-br 4805  df-po 5187  df-so 5188
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator