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Theorem sowlin 4027
Description: A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
Assertion
Ref Expression
sowlin ((𝑅 Or A (B A 𝐶 A 𝐷 A)) → (B𝑅𝐶 → (B𝑅𝐷 𝐷𝑅𝐶)))

Proof of Theorem sowlin
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3737 . . . . 5 (x = B → (x𝑅yB𝑅y))
2 breq1 3737 . . . . . 6 (x = B → (x𝑅zB𝑅z))
32orbi1d 692 . . . . 5 (x = B → ((x𝑅z z𝑅y) ↔ (B𝑅z z𝑅y)))
41, 3imbi12d 223 . . . 4 (x = B → ((x𝑅y → (x𝑅z z𝑅y)) ↔ (B𝑅y → (B𝑅z z𝑅y))))
54imbi2d 219 . . 3 (x = B → ((𝑅 Or A → (x𝑅y → (x𝑅z z𝑅y))) ↔ (𝑅 Or A → (B𝑅y → (B𝑅z z𝑅y)))))
6 breq2 3738 . . . . 5 (y = 𝐶 → (B𝑅yB𝑅𝐶))
7 breq2 3738 . . . . . 6 (y = 𝐶 → (z𝑅yz𝑅𝐶))
87orbi2d 691 . . . . 5 (y = 𝐶 → ((B𝑅z z𝑅y) ↔ (B𝑅z z𝑅𝐶)))
96, 8imbi12d 223 . . . 4 (y = 𝐶 → ((B𝑅y → (B𝑅z z𝑅y)) ↔ (B𝑅𝐶 → (B𝑅z z𝑅𝐶))))
109imbi2d 219 . . 3 (y = 𝐶 → ((𝑅 Or A → (B𝑅y → (B𝑅z z𝑅y))) ↔ (𝑅 Or A → (B𝑅𝐶 → (B𝑅z z𝑅𝐶)))))
11 breq2 3738 . . . . . 6 (z = 𝐷 → (B𝑅zB𝑅𝐷))
12 breq1 3737 . . . . . 6 (z = 𝐷 → (z𝑅𝐶𝐷𝑅𝐶))
1311, 12orbi12d 694 . . . . 5 (z = 𝐷 → ((B𝑅z z𝑅𝐶) ↔ (B𝑅𝐷 𝐷𝑅𝐶)))
1413imbi2d 219 . . . 4 (z = 𝐷 → ((B𝑅𝐶 → (B𝑅z z𝑅𝐶)) ↔ (B𝑅𝐶 → (B𝑅𝐷 𝐷𝑅𝐶))))
1514imbi2d 219 . . 3 (z = 𝐷 → ((𝑅 Or A → (B𝑅𝐶 → (B𝑅z z𝑅𝐶))) ↔ (𝑅 Or A → (B𝑅𝐶 → (B𝑅𝐷 𝐷𝑅𝐶)))))
16 df-iso 4004 . . . . 5 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
17 3anass 875 . . . . . . 7 ((x A y A z A) ↔ (x A (y A z A)))
18 rsp 2343 . . . . . . . . 9 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) → (x Ay A z A (x𝑅y → (x𝑅z z𝑅y))))
19 rsp2 2345 . . . . . . . . 9 (y A z A (x𝑅y → (x𝑅z z𝑅y)) → ((y A z A) → (x𝑅y → (x𝑅z z𝑅y))))
2018, 19syl6 29 . . . . . . . 8 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) → (x A → ((y A z A) → (x𝑅y → (x𝑅z z𝑅y)))))
2120impd 242 . . . . . . 7 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) → ((x A (y A z A)) → (x𝑅y → (x𝑅z z𝑅y))))
2217, 21syl5bi 141 . . . . . 6 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) → ((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y))))
2322adantl 262 . . . . 5 ((𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))) → ((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y))))
2416, 23sylbi 114 . . . 4 (𝑅 Or A → ((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y))))
2524com12 27 . . 3 ((x A y A z A) → (𝑅 Or A → (x𝑅y → (x𝑅z z𝑅y))))
265, 10, 15, 25vtocl3ga 2596 . 2 ((B A 𝐶 A 𝐷 A) → (𝑅 Or A → (B𝑅𝐶 → (B𝑅𝐷 𝐷𝑅𝐶))))
2726impcom 116 1 ((𝑅 Or A (B A 𝐶 A 𝐷 A)) → (B𝑅𝐶 → (B𝑅𝐷 𝐷𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616   w3a 871   = wceq 1226   wcel 1370  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-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-iso 4004
This theorem is referenced by:  sotri2  4645  sotri3  4646  ltsosr  6508  axpre-ltwlin  6576
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