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Theorem soeq1 4022
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 4006 . . 3 (𝑅 = 𝑆 → (𝑅 Po A𝑆 Po A))
2 breq 3736 . . . . . 6 (𝑅 = 𝑆 → (x𝑅yx𝑆y))
3 breq 3736 . . . . . . 7 (𝑅 = 𝑆 → (x𝑅zx𝑆z))
4 breq 3736 . . . . . . 7 (𝑅 = 𝑆 → (z𝑅yz𝑆y))
53, 4orbi12d 694 . . . . . 6 (𝑅 = 𝑆 → ((x𝑅z z𝑅y) ↔ (x𝑆z z𝑆y)))
62, 5imbi12d 223 . . . . 5 (𝑅 = 𝑆 → ((x𝑅y → (x𝑅z z𝑅y)) ↔ (x𝑆y → (x𝑆z z𝑆y))))
762ralbidv 2322 . . . 4 (𝑅 = 𝑆 → (y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ y A z A (x𝑆y → (x𝑆z z𝑆y))))
87ralbidv 2300 . . 3 (𝑅 = 𝑆 → (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ x A y A z A (x𝑆y → (x𝑆z z𝑆y))))
91, 8anbi12d 445 . 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 4004 . 2 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
11 df-iso 4004 . 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 616   = wceq 1226  wral 2280   class class class wbr 3734   Po wpo 4001   Or wor 4002
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-17 1396  ax-ial 1405  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2011  df-clel 2014  df-ral 2285  df-br 3735  df-po 4003  df-iso 4004
This theorem is referenced by: (None)
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