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Theorem soeq1 5612
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 5594 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 5144 . . . . 5 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 biidd 262 . . . . 5 (𝑅 = 𝑆 → (𝑥 = 𝑦𝑥 = 𝑦))
4 breq 5144 . . . . 5 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
52, 3, 43orbi123d 1436 . . . 4 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
652ralbidv 3220 . . 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 5592 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-so 5592 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
107, 8, 93bitr4g 314 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1085   = wceq 1539  wral 3060   class class class wbr 5142   Po wpo 5589   Or wor 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-ex 1779  df-cleq 2728  df-clel 2815  df-ral 3061  df-br 5143  df-po 5591  df-so 5592
This theorem is referenced by:  soeq12d  5614  weeq1  5671  ltsopi  10929  cnso  16284
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