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Theorem soeq1 4116
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 4100 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 3822 . . . . . 6 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 breq 3822 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑧𝑥𝑆𝑧))
4 breq 3822 . . . . . . 7 (𝑅 = 𝑆 → (𝑧𝑅𝑦𝑧𝑆𝑦))
53, 4orbi12d 740 . . . . . 6 (𝑅 = 𝑆 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥𝑆𝑧𝑧𝑆𝑦)))
62, 5imbi12d 232 . . . . 5 (𝑅 = 𝑆 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
762ralbidv 2398 . . . 4 (𝑅 = 𝑆 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
87ralbidv 2376 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
91, 8anbi12d 457 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)))))
10 df-iso 4098 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
11 df-iso 4098 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
129, 10, 113bitr4g 221 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662   = wceq 1287  wral 2355   class class class wbr 3820   Po wpo 4095   Or wor 4096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-cleq 2078  df-clel 2081  df-ral 2360  df-br 3821  df-po 4097  df-iso 4098
This theorem is referenced by: (None)
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