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Theorem iscol2 26196
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 26195 . . 3  |-  ( ph  ->  (coln `  G )  =  { w  |  E. l  e.  (PLines `  G
) w  C_  l } )
32eleq2d 2363 . 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 3212 . . . . 5  |-  ( w  =  A  ->  (
w  C_  l  <->  A  C_  l
) )
65rexbidv 2577 . . . 4  |-  ( w  =  A  ->  ( E. l  e.  (PLines `  G ) w  C_  l 
<->  E. l  e.  (PLines `  G ) A  C_  l ) )
76elabg 2928 . . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165   ` cfv 5271  PLinescplines 26161  Igcig 26163  colnccol 26193
This theorem is referenced by:  iscol3  26197  isibg1a6  26228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-col 26194
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