Theorem pocl 3596

Description: Properties of partial order relation in class notation.

Assertion

Ref	Expression
pocl	|- (R Po A -> ((B e. A /\ C e. A /\ D e. A) -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))

Proof of Theorem pocl

Step	Hyp	Ref	Expression
1		id 73	. . . . . . 7 |- (x = B -> x = B)
2	1, 1	breq12d 3351	. . . . . 6 |- (x = B -> (xRx <-> BRB))
3	2	notbid 673	. . . . 5 |- (x = B -> (-. xRx <-> -. BRB))
4		breq1 3341	. . . . . . 7 |- (x = B -> (xRy <-> BRy))
5	4	anbi1d 679	. . . . . 6 |- (x = B -> ((xRy /\ yRz) <-> (BRy /\ yRz)))
6		breq1 3341	. . . . . 6 |- (x = B -> (xRz <-> BRz))
7	5, 6	imbi12d 688	. . . . 5 |- (x = B -> (((xRy /\ yRz) -> xRz) <-> ((BRy /\ yRz) -> BRz)))
8	3, 7	anbi12d 690	. . . 4 |- (x = B -> ((-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> (-. BRB /\ ((BRy /\ yRz) -> BRz))))
9	8	imbi2d 674	. . 3 |- (x = B -> ((R Po A -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) <-> (R Po A -> (-. BRB /\ ((BRy /\ yRz) -> BRz)))))
10		breq2 3342	. . . . . . 7 |- (y = C -> (BRy <-> BRC))
11		breq1 3341	. . . . . . 7 |- (y = C -> (yRz <-> CRz))
12	10, 11	anbi12d 690	. . . . . 6 |- (y = C -> ((BRy /\ yRz) <-> (BRC /\ CRz)))
13	12	imbi1d 675	. . . . 5 |- (y = C -> (((BRy /\ yRz) -> BRz) <-> ((BRC /\ CRz) -> BRz)))
14	13	anbi2d 678	. . . 4 |- (y = C -> ((-. BRB /\ ((BRy /\ yRz) -> BRz)) <-> (-. BRB /\ ((BRC /\ CRz) -> BRz))))
15	14	imbi2d 674	. . 3 |- (y = C -> ((R Po A -> (-. BRB /\ ((BRy /\ yRz) -> BRz))) <-> (R Po A -> (-. BRB /\ ((BRC /\ CRz) -> BRz)))))
16		breq2 3342	. . . . . . 7 |- (z = D -> (CRz <-> CRD))
17	16	anbi2d 678	. . . . . 6 |- (z = D -> ((BRC /\ CRz) <-> (BRC /\ CRD)))
18		breq2 3342	. . . . . 6 |- (z = D -> (BRz <-> BRD))
19	17, 18	imbi12d 688	. . . . 5 |- (z = D -> (((BRC /\ CRz) -> BRz) <-> ((BRC /\ CRD) -> BRD)))
20	19	anbi2d 678	. . . 4 |- (z = D -> ((-. BRB /\ ((BRC /\ CRz) -> BRz)) <-> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
21	20	imbi2d 674	. . 3 |- (z = D -> ((R Po A -> (-. BRB /\ ((BRC /\ CRz) -> BRz))) <-> (R Po A -> (-. BRB /\ ((BRC /\ CRD) -> BRD)))))
22		df-po 3591	. . . . . . . 8 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
23		r3al 2151	. . . . . . . 8 |- (A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
24	22, 23	bitri 190	. . . . . . 7 |- (R Po A <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
25	24	biimpi 168	. . . . . 6 |- (R Po A -> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
26	25	19.21bbi 1409	. . . . 5 |- (R Po A -> A.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
27	26	19.21bi 1408	. . . 4 |- (R Po A -> ((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
28	27	com12 14	. . 3 |- ((x e. A /\ y e. A /\ z e. A) -> (R Po A -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
29	9, 15, 21, 28	vtocl3ga 2354	. 2 |- ((B e. A /\ C e. A /\ D e. A) -> (R Po A -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
30	29	com12 14	1 |- (R Po A -> ((B e. A /\ C e. A /\ D e. A) -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338   Po wpo 3589

This theorem is referenced by:  poirr 3597  potr 3598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-po 3591

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