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Theorem isibg1spa 26123
 Description: If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isibg1a3a.1 PPoints
isibg1a3a.2 btw
isibg1a3a.3 Ibg
isibg1a3a.4
isibg1a3a.5
isibg1a3a.6
isibg1a3a.7
Assertion
Ref Expression
isibg1spa

Proof of Theorem isibg1spa
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg1a3a.3 . 2 Ibg
2 isibg1a3a.1 . . . 4 PPoints
3 eqid 2283 . . . 4 PLines PLines
4 isibg1a3a.2 . . . 4 btw
5 eqid 2283 . . . 4 coln coln
62, 3, 4, 5isibg2 26110 . . 3 Ibg Ig coln coln PLines
7 isibg1a3a.4 . . . . . . 7
8 isibg1a3a.5 . . . . . . 7
92, 3isibg2aalem1 26113 . . . . . . . . 9 coln coln PLines coln coln PLines
10 isibg1a3a.6 . . . . . . . . . . . 12
11 isibg2aalem2 26114 . . . . . . . . . . . . . 14 coln coln PLines coln coln PLines
12 isibg1a3a.7 . . . . . . . . . . . . . . . 16
13 simp32 992 . . . . . . . . . . . . . . . 16 coln
1412, 13imim12i 53 . . . . . . . . . . . . . . 15 coln
1514ad2antrl 708 . . . . . . . . . . . . . 14 coln coln PLines
1611, 15syl6 29 . . . . . . . . . . . . 13 coln coln PLines
1716com23 72 . . . . . . . . . . . 12 coln coln PLines
1810, 17mpcom 32 . . . . . . . . . . 11 coln coln PLines
1918com12 27 . . . . . . . . . 10 coln coln PLines
2019adantl 452 . . . . . . . . 9 coln coln PLines
219, 20syl6 29 . . . . . . . 8 coln coln PLines
2221com23 72 . . . . . . 7 coln coln PLines
237, 8, 22syl2anc 642 . . . . . 6 coln coln PLines
2423pm2.43i 43 . . . . 5 coln coln PLines
2524com12 27 . . . 4 coln coln PLines
2625adantl 452 . . 3 Ig coln coln PLines
276, 26sylbi 187 . 2 Ibg
281, 27mpcom 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:  isibg1a6  26125  lppotos  26144 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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