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Theorem isibg1a 26214
 Description: An incidence-betweenness geometry is an incidence geometry. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Hypothesis
Ref Expression
isibg1a.1 Ibg
Assertion
Ref Expression
isibg1a Ig

Proof of Theorem isibg1a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg1a.1 . 2 Ibg
2 eqid 2296 . . . 4 PPoints PPoints
3 eqid 2296 . . . 4 PLines PLines
4 eqid 2296 . . . 4 btw btw
5 eqid 2296 . . . 4 coln coln
62, 3, 4, 5isibg2 26213 . . 3 Ibg Ig PPoints PPoints PPoints PPoints PPoints btw btw btw PPoints coln btw btw btw btw btw btw btw btw btw btw btw coln PLines btw btw btw btw btw btw
76simplbi 446 . 2 Ibg Ig
81, 7syl 15 1 Ig
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3o 933   w3a 934   wceq 1632   wcel 1696   wne 2459   wnel 2460  wral 2556  wrex 2557   cin 3164  c0 3468  ctp 3655  cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163  colnccol 26193  btwcbtw 26209  Ibgcibg 26210 This theorem is referenced by:  isibg1a6  26228  isibg1a7  26229  isibg1a8  26230  segline  26244  lppotos  26247  bsstrs  26249  rayline  26259  hpd  26272  abhp  26276  bhp3  26280 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-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  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-ov 5877  df-ibg2 26212
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