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Theorem soeq1 4246
 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 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poeq1 4230 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 3940 . . . . . 6 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 breq 3940 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑧𝑥𝑆𝑧))
4 breq 3940 . . . . . . 7 (𝑅 = 𝑆 → (𝑧𝑅𝑦𝑧𝑆𝑦))
53, 4orbi12d 783 . . . . . 6 (𝑅 = 𝑆 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥𝑆𝑧𝑧𝑆𝑦)))
62, 5imbi12d 233 . . . . 5 (𝑅 = 𝑆 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
762ralbidv 2463 . . . 4 (𝑅 = 𝑆 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
87ralbidv 2439 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
91, 8anbi12d 465 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)))))
10 df-iso 4228 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
11 df-iso 4228 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
129, 10, 113bitr4g 222 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332  ∀wral 2417   class class class wbr 3938   Po wpo 4225   Or wor 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-ral 2422  df-br 3939  df-po 4227  df-iso 4228 This theorem is referenced by: (None)
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