Theorem pocl 2839

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 59	. . . . . . 7 |- (x = B -> x = B)
2	1, 1	breq12d 2626	. . . . . 6 |- (x = B -> (xRx <-> BRB))
3	2	negbid 610	. . . . 5 |- (x = B -> (-. xRx <-> -. BRB))
4		breq1 2617	. . . . . . 7 |- (x = B -> (xRy <-> BRy))
5	4	anbi1d 616	. . . . . 6 |- (x = B -> ((xRy /\ yRz) <-> (BRy /\ yRz)))
6		breq1 2617	. . . . . 6 |- (x = B -> (xRz <-> BRz))
7	5, 6	imbi12d 625	. . . . 5 |- (x = B -> (((xRy /\ yRz) -> xRz) <-> ((BRy /\ yRz) -> BRz)))
8	3, 7	anbi12d 627	. . . 4 |- (x = B -> ((-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> (-. BRB /\ ((BRy /\ yRz) -> BRz))))
9	8	imbi2d 611	. . 3 |- (x = B -> ((R Po A -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) <-> (R Po A -> (-. BRB /\ ((BRy /\ yRz) -> BRz)))))
10		breq2 2618	. . . . . . 7 |- (y = C -> (BRy <-> BRC))
11		breq1 2617	. . . . . . 7 |- (y = C -> (yRz <-> CRz))
12	10, 11	anbi12d 627	. . . . . 6 |- (y = C -> ((BRy /\ yRz) <-> (BRC /\ CRz)))
13	12	imbi1d 612	. . . . 5 |- (y = C -> (((BRy /\ yRz) -> BRz) <-> ((BRC /\ CRz) -> BRz)))
14	13	anbi2d 615	. . . 4 |- (y = C -> ((-. BRB /\ ((BRy /\ yRz) -> BRz)) <-> (-. BRB /\ ((BRC /\ CRz) -> BRz))))
15	14	imbi2d 611	. . 3 |- (y = C -> ((R Po A -> (-. BRB /\ ((BRy /\ yRz) -> BRz))) <-> (R Po A -> (-. BRB /\ ((BRC /\ CRz) -> BRz)))))
16		breq2 2618	. . . . . . 7 |- (z = D -> (CRz <-> CRD))
17	16	anbi2d 615	. . . . . 6 |- (z = D -> ((BRC /\ CRz) <-> (BRC /\ CRD)))
18		breq2 2618	. . . . . 6 |- (z = D -> (BRz <-> BRD))
19	17, 18	imbi12d 625	. . . . 5 |- (z = D -> (((BRC /\ CRz) -> BRz) <-> ((BRC /\ CRD) -> BRD)))
20	19	anbi2d 615	. . . 4 |- (z = D -> ((-. BRB /\ ((BRC /\ CRz) -> BRz)) <-> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
21	20	imbi2d 611	. . 3 |- (z = D -> ((R Po A -> (-. BRB /\ ((BRC /\ CRz) -> BRz))) <-> (R Po A -> (-. BRB /\ ((BRC /\ CRD) -> BRD)))))
22		df-po 2835	. . . . . . . 8 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
23		r3al 1687	. . . . . . . 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	bitr 173	. . . . . . 7 |- (R Po A <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
25	24	biimp 151	. . . . . 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 1059	. . . . 5 |- (R Po A -> A.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
27	26	19.21bi 1058	. . . 4 |- (R Po A -> ((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
28	27	com12 11	. . 3 |- ((x e. A /\ y e. A /\ z e. A) -> (R Po A -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
29	9, 15, 21, 28	vtocl3ga 1850	. 2 |- ((B e. A /\ C e. A /\ D e. A) -> (R Po A -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
30	29	com12 11	1 |- (R Po A -> ((B e. A /\ C e. A /\ D e. A) -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774  A.wal 952   = wceq 954   e. wcel 956  A.wral 1642   class class class wbr 2614   Po wpo 2833

This theorem is referenced by:  poirr 2840  potr 2841

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-po 2835

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