brcolinear - Mathbox for Scott Fenton

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Theorem brcolinear 32166

Description: The binary relation form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)

**Assertion**
Ref	Expression
brcolinear	$/\$ $/\$ $/\$ $\/$ $\/$

Proof of Theorem brcolinear Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcolinear2 32165	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
2	1	3adant1 1079	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
3	2	adantl 482	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
4		simpr 477	. . . 4 $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
5	4	rexlimivw 3029	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
6		fveq2 6191	. . . . . . . 8
7	6	eleq2d 2687	. . . . . . 7
8	6	eleq2d 2687	. . . . . . 7
9	6	eleq2d 2687	. . . . . . 7
10	7, 8, 9	3anbi123d 1399	. . . . . 6 $/\$ $/\$ $/\$ $/\$
11	10	anbi1d 741	. . . . 5 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
12	11	rspcev 3309	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
13	12	expr 643	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $\/$
14	5, 13	impbid2 216	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
15	3, 14	bitrd 268	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

cop 4183 class class class wbr 4653

cfv 5888

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-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-iota 5851 df-fv 5896 df-oprab 6654 df-colinear 32146

This theorem is referenced by: colinearperm1 32169 colinearperm3 32170 colineartriv1 32174 colineartriv2 32175 btwncolinear1 32176 colinearxfr 32182 lineext 32183 fscgr 32187 colinbtwnle 32225 broutsideof2 32229 lineunray 32254 lineelsb2 32255