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Theorem iscol2 26093
Description: The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
iscola2.1  |-  ( ph  ->  G  e. Ig )
iscola2.2  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
iscol2  |-  ( ph  ->  ( A  e.  (coln `  G )  <->  E. l  e.  (PLines `  G ) A  C_  l ) )
Distinct variable groups:    A, l    G, l
Allowed substitution hints:    ph( l)    B( l)

Proof of Theorem iscol2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 iscola2.1 . . . 4  |-  ( ph  ->  G  e. Ig )
21iscola2 26092 . . 3  |-  ( ph  ->  (coln `  G )  =  { w  |  E. l  e.  (PLines `  G
) w  C_  l } )
32eleq2d 2350 . 2  |-  ( ph  ->  ( A  e.  (coln `  G )  <->  A  e.  { w  |  E. l  e.  (PLines `  G )
w  C_  l }
) )
4 iscola2.2 . . 3  |-  ( ph  ->  A  e.  B )
5 sseq1 3199 . . . . 5  |-  ( w  =  A  ->  (
w  C_  l  <->  A  C_  l
) )
65rexbidv 2564 . . . 4  |-  ( w  =  A  ->  ( E. l  e.  (PLines `  G ) w  C_  l 
<->  E. l  e.  (PLines `  G ) A  C_  l ) )
76elabg 2915 . . 3  |-  ( A  e.  B  ->  ( A  e.  { w  |  E. l  e.  (PLines `  G ) w  C_  l }  <->  E. l  e.  (PLines `  G ) A  C_  l ) )
84, 7syl 15 . 2  |-  ( ph  ->  ( A  e.  {
w  |  E. l  e.  (PLines `  G )
w  C_  l }  <->  E. l  e.  (PLines `  G ) A  C_  l ) )
93, 8bitrd 244 1  |-  ( ph  ->  ( A  e.  (coln `  G )  <->  E. l  e.  (PLines `  G ) A  C_  l ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   ` cfv 5255  PLinescplines 26058  Igcig 26060  colnccol 26090
This theorem is referenced by:  iscol3  26094  isibg1a6  26125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-col 26091
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