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Theorem isig22 26168
 Description: There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Hypotheses
Ref Expression
isig.1 PPoints
isig.2 PLines
isig22.1 Ig
Assertion
Ref Expression
isig22
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem isig22
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isig22.1 . 2 Ig
2 isig.1 . . . 4 PPoints
3 isig.2 . . . 4 PLines
42, 3bisig0 26165 . . 3 Ig
5 simp22 989 . . 3
64, 5sylbi 187 . 2 Ig
71, 6syl 15 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  wreu 2558  cvv 2801   wss 3165  cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163 This theorem is referenced by:  isig2a2  26169 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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