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Theorem soeq1 5083
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))

Proof of Theorem soeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poeq1 5067 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 4687 . . . . 5 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 biidd 252 . . . . 5 (𝑅 = 𝑆 → (𝑥 = 𝑦𝑥 = 𝑦))
4 breq 4687 . . . . 5 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
52, 3, 43orbi123d 1438 . . . 4 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
652ralbidv 3018 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
71, 6anbi12d 747 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥))))
8 df-so 5065 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-so 5065 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
107, 8, 93bitr4g 303 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1053   = wceq 1523  wral 2941   class class class wbr 4685   Po wpo 5062   Or wor 5063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-ex 1745  df-cleq 2644  df-clel 2647  df-ral 2946  df-br 4686  df-po 5064  df-so 5065
This theorem is referenced by:  weeq1  5131  ltsopi  9748  cnso  15020  opsrtoslem2  19533  soeq12d  37925
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