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Theorem tethpnc2 26174
 Description: There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
isig.1 PPoints
isig.2 PLines
tethpnc2.1 Ig
tethpnc2.2
Assertion
Ref Expression
tethpnc2
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem tethpnc2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isig.1 . . 3 PPoints
2 isig.2 . . 3 PLines
3 tethpnc2.1 . . 3 Ig
41, 2, 3tethpnc 26173 . 2
5 tethpnc2.2 . . . . . . 7
6 eleq2 2357 . . . . . . . . . 10
7 eleq2 2357 . . . . . . . . . 10
8 eleq2 2357 . . . . . . . . . 10
96, 7, 83anbi123d 1252 . . . . . . . . 9
109notbid 285 . . . . . . . 8
1110rspcv 2893 . . . . . . 7
125, 11syl 15 . . . . . 6
1312anim2d 548 . . . . 5
1413reximdv 2667 . . . 4
1514reximdv 2667 . . 3
1615reximdv 2667 . 2
174, 16mpd 14 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696   wne 2459  wral 2556  wrex 2557  cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163 This theorem is referenced by:  gltpntl  26175 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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165 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-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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ig2 26164
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