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Theorem soeq1 4077
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 4061 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 3791 . . . . . 6 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 breq 3791 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑧𝑥𝑆𝑧))
4 breq 3791 . . . . . . 7 (𝑅 = 𝑆 → (𝑧𝑅𝑦𝑧𝑆𝑦))
53, 4orbi12d 715 . . . . . 6 (𝑅 = 𝑆 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥𝑆𝑧𝑧𝑆𝑦)))
62, 5imbi12d 227 . . . . 5 (𝑅 = 𝑆 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
762ralbidv 2363 . . . 4 (𝑅 = 𝑆 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
87ralbidv 2341 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
91, 8anbi12d 450 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)))))
10 df-iso 4059 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
11 df-iso 4059 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
129, 10, 113bitr4g 216 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 637   = wceq 1257  wral 2321   class class class wbr 3789   Po wpo 4056   Or wor 4057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-4 1414  ax-17 1433  ax-ial 1441  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-cleq 2047  df-clel 2050  df-ral 2326  df-br 3790  df-po 4058  df-iso 4059
This theorem is referenced by: (None)
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