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Theorem elhalop2 26172
 Description: Every line has at least one point. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
Hypotheses
Ref Expression
isig.1 PPoints
isig.2 PLines
elhalop2.1 Ig
elhalop2.2
Assertion
Ref Expression
elhalop2
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem elhalop2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isig.1 . . 3 PPoints
2 isig.2 . . 3 PLines
3 elhalop2.1 . . 3 Ig
4 elhalop2.2 . . 3
51, 2, 3, 4elhaltdp2 26171 . 2
6 simp2 956 . . . 4
76rexlimivw 2676 . . 3
87reximi 2663 . 2
95, 8syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 934   wceq 1632   wcel 1696   wne 2459  wrex 2557  cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163 This theorem is referenced by:  aline  26177  lppotos  26247 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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