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Theorem soeq1 5252
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 5236 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 4845 . . . . 5 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 biidd 254 . . . . 5 (𝑅 = 𝑆 → (𝑥 = 𝑦𝑥 = 𝑦))
4 breq 4845 . . . . 5 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
52, 3, 43orbi123d 1560 . . . 4 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
652ralbidv 3170 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
71, 6anbi12d 625 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥))))
8 df-so 5234 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-so 5234 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
107, 8, 93bitr4g 306 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3o 1107   = wceq 1653  wral 3089   class class class wbr 4843   Po wpo 5231   Or wor 5232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-ex 1876  df-cleq 2792  df-clel 2795  df-ral 3094  df-br 4844  df-po 5233  df-so 5234
This theorem is referenced by:  weeq1  5300  ltsopi  9998  cnso  15312  opsrtoslem2  19807  soeq12d  38393
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