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Theorem isig2a2 26169
 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
isig2a2.1 Ig
isig2a2.2
isig2a2.3
isig2a2.4
Assertion
Ref Expression
isig2a2
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem isig2a2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isig2a2.4 . 2
2 isig2a2.2 . . 3
3 isig2a2.3 . . 3
4 isig.1 . . . 4 PPoints
5 isig.2 . . . 4 PLines
6 isig2a2.1 . . . 4 Ig
74, 5, 6isig22 26168 . . 3
8 neeq1 2467 . . . . 5
9 eleq1 2356 . . . . . . 7
109anbi1d 685 . . . . . 6
1110reubidv 2737 . . . . 5
128, 11imbi12d 311 . . . 4
13 neeq2 2468 . . . . 5
14 eleq1 2356 . . . . . . 7
1514anbi2d 684 . . . . . 6
1615reubidv 2737 . . . . 5
1713, 16imbi12d 311 . . . 4
1812, 17rspc2va 2904 . . 3
192, 3, 7, 18syl21anc 1181 . 2
201, 19mpd 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696   wne 2459  wral 2556  wreu 2558  cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163 This theorem is referenced by:  lineval42  26183  lineval12  26184  lineval22  26185 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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