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Theorem soeq1 5470
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 5453 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 5044 . . . . 5 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 biidd 264 . . . . 5 (𝑅 = 𝑆 → (𝑥 = 𝑦𝑥 = 𝑦))
4 breq 5044 . . . . 5 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
52, 3, 43orbi123d 1431 . . . 4 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
652ralbidv 3186 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
71, 6anbi12d 632 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥))))
8 df-so 5451 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-so 5451 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
107, 8, 93bitr4g 316 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3o 1082   = wceq 1537  wral 3125   class class class wbr 5042   Po wpo 5448   Or wor 5449
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-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-ex 1781  df-cleq 2813  df-clel 2891  df-ral 3130  df-br 5043  df-po 5450  df-so 5451
This theorem is referenced by:  weeq1  5519  ltsopi  10288  cnso  15580  opsrtoslem2  20241  soeq12d  39775
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