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Theorem soeq1 4043
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 4027 . . 3 (𝑅 = 𝑆 → (𝑅 Po A𝑆 Po A))
2 breq 3757 . . . . . 6 (𝑅 = 𝑆 → (x𝑅yx𝑆y))
3 breq 3757 . . . . . . 7 (𝑅 = 𝑆 → (x𝑅zx𝑆z))
4 breq 3757 . . . . . . 7 (𝑅 = 𝑆 → (z𝑅yz𝑆y))
53, 4orbi12d 706 . . . . . 6 (𝑅 = 𝑆 → ((x𝑅z z𝑅y) ↔ (x𝑆z z𝑆y)))
62, 5imbi12d 223 . . . . 5 (𝑅 = 𝑆 → ((x𝑅y → (x𝑅z z𝑅y)) ↔ (x𝑆y → (x𝑆z z𝑆y))))
762ralbidv 2342 . . . 4 (𝑅 = 𝑆 → (y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ y A z A (x𝑆y → (x𝑆z z𝑆y))))
87ralbidv 2320 . . 3 (𝑅 = 𝑆 → (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ x A y A z A (x𝑆y → (x𝑆z z𝑆y))))
91, 8anbi12d 442 . 2 (𝑅 = 𝑆 → ((𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))) ↔ (𝑆 Po A x A y A z A (x𝑆y → (x𝑆z z𝑆y)))))
10 df-iso 4025 . 2 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
11 df-iso 4025 . 2 (𝑆 Or A ↔ (𝑆 Po A x A y A z A (x𝑆y → (x𝑆z z𝑆y))))
129, 10, 113bitr4g 212 1 (𝑅 = 𝑆 → (𝑅 Or A𝑆 Or A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   = wceq 1242  wral 2300   class class class wbr 3755   Po wpo 4022   Or wor 4023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-ral 2305  df-br 3756  df-po 4024  df-iso 4025
This theorem is referenced by: (None)
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