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Theorem sowlin 4048
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 3758 . . . . 5 (x = B → (x𝑅yB𝑅y))
2 breq1 3758 . . . . . 6 (x = B → (x𝑅zB𝑅z))
32orbi1d 704 . . . . 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 3759 . . . . 5 (y = 𝐶 → (B𝑅yB𝑅𝐶))
7 breq2 3759 . . . . . 6 (y = 𝐶 → (z𝑅yz𝑅𝐶))
87orbi2d 703 . . . . 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 3759 . . . . . 6 (z = 𝐷 → (B𝑅zB𝑅𝐷))
12 breq1 3758 . . . . . 6 (z = 𝐷 → (z𝑅𝐶𝐷𝑅𝐶))
1311, 12orbi12d 706 . . . . 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 4025 . . . . 5 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
17 3anass 888 . . . . . . 7 ((x A y A z A) ↔ (x A (y A z A)))
18 rsp 2363 . . . . . . . . 9 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) → (x Ay A z A (x𝑅y → (x𝑅z z𝑅y))))
19 rsp2 2365 . . . . . . . . 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 2617 . 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 628   w3a 884   = wceq 1242   wcel 1390  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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-iso 4025
This theorem is referenced by:  sotri2  4665  sotri3  4666  addextpr  6592  cauappcvgprlemloc  6623  ltsosr  6672  axpre-ltwlin  6747  xrlelttr  8472  xrltletr  8473  xrletr  8474
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