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 e. A /\ C e. A /\ D e. A)) -> (B R C -> (B R D \/ D R C)))

Proof of Theorem sowlin

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 3758	. . . . 5 |- (x = B -> (x Ry <-> B Ry))
2		breq1 3758	. . . . . 6 |- (x = B -> (x Rz <-> B Rz))
3	2	orbi1d 704	. . . . 5 |- (x = B -> ((x Rz \/ z Ry) <-> (B Rz \/ z Ry)))
4	1, 3	imbi12d 223	. . . 4 |- (x = B -> ((x Ry -> (x Rz \/ z Ry)) <-> (B Ry -> (B Rz \/ z Ry))))
5	4	imbi2d 219	. . 3 |- (x = B -> (( R Or A -> (x Ry -> (x Rz \/ z Ry))) <-> ( R Or A -> (B Ry -> (B Rz \/ z Ry)))))
6		breq2 3759	. . . . 5 |- (y = C -> (B Ry <-> B R C))
7		breq2 3759	. . . . . 6 |- (y = C -> (z Ry <-> z R C))
8	7	orbi2d 703	. . . . 5 |- (y = C -> ((B Rz \/ z Ry) <-> (B Rz \/ z R C)))
9	6, 8	imbi12d 223	. . . 4 |- (y = C -> ((B Ry -> (B Rz \/ z Ry)) <-> (B R C -> (B Rz \/ z R C))))
10	9	imbi2d 219	. . 3 |- (y = C -> (( R Or A -> (B Ry -> (B Rz \/ z Ry))) <-> ( R Or A -> (B R C -> (B Rz \/ z R C)))))
11		breq2 3759	. . . . . 6 |- (z = D -> (B Rz <-> B R D))
12		breq1 3758	. . . . . 6 |- (z = D -> (z R C <-> D R C))
13	11, 12	orbi12d 706	. . . . 5 |- (z = D -> ((B Rz \/ z R C) <-> (B R D \/ D R C)))
14	13	imbi2d 219	. . . 4 |- (z = D -> ((B R C -> (B Rz \/ z R C)) <-> (B R C -> (B R D \/ D R C))))
15	14	imbi2d 219	. . 3 |- (z = D -> (( R Or A -> (B R C -> (B Rz \/ z R C))) <-> ( R Or A -> (B R C -> (B R D \/ D R C)))))
16		df-iso 4025	. . . . 5 |- ( R Or A <-> ( R Po A /\ A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry))))
17		3anass 888	. . . . . . 7 |- ((x e. A /\ y e. A /\ z e. A) <-> (x e. A /\ (y e. A /\ z e. A)))
18		rsp 2363	. . . . . . . . 9 |- (A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) -> (x e. A -> A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry))))
19		rsp2 2365	. . . . . . . . 9 |- (A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) -> ((y e. A /\ z e. A) -> (x Ry -> (x Rz \/ z Ry))))
20	18, 19	syl6 29	. . . . . . . 8 |- (A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) -> (x e. A -> ((y e. A /\ z e. A) -> (x Ry -> (x Rz \/ z Ry)))))
21	20	impd 242	. . . . . . 7 |- (A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) -> ((x e. A /\ (y e. A /\ z e. A)) -> (x Ry -> (x Rz \/ z Ry))))
22	17, 21	syl5bi 141	. . . . . 6 |- (A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) -> ((x e. A /\ y e. A /\ z e. A) -> (x Ry -> (x Rz \/ z Ry))))
23	22	adantl 262	. . . . 5 |- (( R Po A /\ A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry))) -> ((x e. A /\ y e. A /\ z e. A) -> (x Ry -> (x Rz \/ z Ry))))
24	16, 23	sylbi 114	. . . 4 |- ( R Or A -> ((x e. A /\ y e. A /\ z e. A) -> (x Ry -> (x Rz \/ z Ry))))
25	24	com12 27	. . 3 |- ((x e. A /\ y e. A /\ z e. A) -> ( R Or A -> (x Ry -> (x Rz \/ z Ry))))
26	5, 10, 15, 25	vtocl3ga 2617	. 2 |- ((B e. A /\ C e. A /\ D e. A) -> ( R Or A -> (B R C -> (B R D \/ D R C))))
27	26	impcom 116	1 |- (( R Or A /\ (B e. A /\ C e. A /\ D e. A)) -> (B R C -> (B R D \/ D R C)))

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 628   /\ w3a 884   = wceq 1242   e. wcel 1390  A.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-bndl 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  6593  cauappcvgprlemloc  6624  caucvgprlemloc  6646  ltsosr  6692  axpre-ltwlin  6767  xrlelttr  8492  xrltletr  8493  xrletr  8494