brcolinear2 - Mathbox for Scott Fenton

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Theorem brcolinear2 32165

Description: Alternate colinearity binary relation. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
brcolinear2	$/\$ $/\$ $/\$ $/\$ $\/$ $\/$

Distinct variable groups:

Allowed substitution hints:

(

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Proof of Theorem brcolinear2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		colinrel 32164	. . . 4
2	1	brrelexi 5158	. . 3
3	2	a1i 11	. 2 $/\$
4		elex 3212	. . . . . 6
5	4	3ad2ant1 1082	. . . . 5 $/\$ $/\$
6	5	adantr 481	. . . 4 $/\$ $/\$ $/\$ $\/$ $\/$
7	6	rexlimivw 3029	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
8	7	a1i 11	. 2 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
9		df-br 4654	. . . . . 6
10		df-colinear 32146	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $\/$
11	10	eleq2i 2693	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $\/$
12	9, 11	bitri 264	. . . . 5 $/\$ $/\$ $/\$ $\/$ $\/$
13		opex 4932	. . . . . . 7
14		opelcnvg 5302	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
15	13, 14	mpan2 707	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
16	15	3ad2ant3 1084	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
17	12, 16	syl5bb 272	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
18		eleq1 2689	. . . . . . . 8
19	18	3anbi2d 1404	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
20		opeq1 4402	. . . . . . . . 9
21	20	breq2d 4665	. . . . . . . 8
22		breq1 4656	. . . . . . . 8
23		opeq2 4403	. . . . . . . . 9
24	23	breq2d 4665	. . . . . . . 8
25	21, 22, 24	3orbi123d 1398	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
26	19, 25	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
27	26	rexbidv 3052	. . . . 5 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
28		eleq1 2689	. . . . . . . 8
29	28	3anbi3d 1405	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
30		opeq2 4403	. . . . . . . . 9
31	30	breq2d 4665	. . . . . . . 8
32		opeq1 4402	. . . . . . . . 9
33	32	breq2d 4665	. . . . . . . 8
34		breq1 4656	. . . . . . . 8
35	31, 33, 34	3orbi123d 1398	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
36	29, 35	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
37	36	rexbidv 3052	. . . . 5 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
38		eleq1 2689	. . . . . . . 8
39	38	3anbi1d 1403	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
40		breq1 4656	. . . . . . . 8
41		opeq2 4403	. . . . . . . . 9
42	41	breq2d 4665	. . . . . . . 8
43		opeq1 4402	. . . . . . . . 9
44	43	breq2d 4665	. . . . . . . 8
45	40, 42, 44	3orbi123d 1398	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
46	39, 45	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
47	46	rexbidv 3052	. . . . 5 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
48	27, 37, 47	eloprabg 6748	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
49	17, 48	bitrd 268	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
50	49	3expia 1267	. 2 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
51	3, 8, 50	pm5.21ndd 369	1 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $\/$ w3o 1036 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913

cvv 3200

cop 4183 class class class wbr 4653

ccnv 5113

cfv 5888

coprab 6651

cn 11020

cee 25768

cbtwn 25769

ccolin 32144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-oprab 6654 df-colinear 32146

This theorem is referenced by: brcolinear 32166

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