Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcolinear Structured version   Unicode version

Theorem brcolinear 25995
 Description: The binary relationship 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.
StepHypRef Expression
1 brcolinear2 25994 . . . 4
4 simpr 449 . . . 4
54rexlimivw 2828 . . 3
6 fveq2 5730 . . . . . . . 8
76eleq2d 2505 . . . . . . 7
86eleq2d 2505 . . . . . . 7
96eleq2d 2505 . . . . . . 7
107, 8, 93anbi123d 1255 . . . . . 6
1110anbi1d 687 . . . . 5
1211rspcev 3054 . . . 4
1312expr 600 . . 3
145, 13impbid2 197 . 2
153, 14bitrd 246 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3o 936   w3a 937   wceq 1653   wcel 1726  wrex 2708  cop 3819   class class class wbr 4214  cfv 5456  cn 10002  cee 25829   cbtwn 25830   ccolin 25973 This theorem is referenced by:  colinearperm1  25998  colinearperm3  25999  colineartriv1  26003  colineartriv2  26004  btwncolinear1  26005  colinearxfr  26011  lineext  26012  fscgr  26016  colinbtwnle  26054  broutsideof2  26058  lineunray  26083  lineelsb2  26084 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-iota 5420  df-fv 5464  df-oprab 6087  df-colinear 25977
 Copyright terms: Public domain W3C validator