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Theorem isibg2a 26118
 Description: Two distinct points have a point before, between and after them. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
Hypotheses
Ref Expression
isibg2a.1 PPoints
isibg2a.2 btw
isibg2a.3 Ibg
isibg2a.4
isibg2a.5
isibg2a.6
Assertion
Ref Expression
isibg2a
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem isibg2a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg2a.3 . 2 Ibg
2 isibg2a.1 . . . 4 PPoints
3 eqid 2283 . . . 4 PLines PLines
4 isibg2a.2 . . . 4 btw
5 eqid 2283 . . . 4 coln coln
62, 3, 4, 5isibg2 26110 . . 3 Ibg Ig coln coln PLines
7 isibg2a.4 . . . . . . 7
8 isibg2a.5 . . . . . . 7
92, 3isibg2aalem1 26113 . . . . . . . . 9 coln coln PLines coln coln PLines
10 isibg2a.6 . . . . . . . . . . 11
1110imim1i 54 . . . . . . . . . 10
1211adantr 451 . . . . . . . . 9 coln coln PLines
139, 12syl6 29 . . . . . . . 8 coln coln PLines
1413com23 72 . . . . . . 7 coln coln PLines
157, 8, 14syl2anc 642 . . . . . 6 coln coln PLines
1615pm2.43i 43 . . . . 5 coln coln PLines
1716com12 27 . . . 4 coln coln PLines
1817adantl 452 . . 3 Ig coln coln PLines
196, 18sylbi 187 . 2 Ibg
201, 19mpcom 32 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3o 933   w3a 934   wceq 1623   wcel 1684   wne 2446   wnel 2447  wral 2543  wrex 2544   cin 3151  c0 3455  ctp 3642  cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060  colnccol 26090  btwcbtw 26106  Ibgcibg 26107 This theorem is referenced by:  isibg2a1  26119  isibg2a2  26120  isibg2a3  26121 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149 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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-ibg2 26109
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