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Theorem tgbtwncomb 24005
Description: Betweeness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
tkgeom.p  |-  P  =  ( Base `  G
)
tkgeom.d  |-  .-  =  ( dist `  G )
tkgeom.i  |-  I  =  (Itv `  G )
tkgeom.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwntriv2.1  |-  ( ph  ->  A  e.  P )
tgbtwntriv2.2  |-  ( ph  ->  B  e.  P )
tgbtwncomb.3  |-  ( ph  ->  C  e.  P )
Assertion
Ref Expression
tgbtwncomb  |-  ( ph  ->  ( B  e.  ( A I C )  <-> 
B  e.  ( C I A ) ) )

Proof of Theorem tgbtwncomb
StepHypRef Expression
1 tkgeom.p . . 3  |-  P  =  ( Base `  G
)
2 tkgeom.d . . 3  |-  .-  =  ( dist `  G )
3 tkgeom.i . . 3  |-  I  =  (Itv `  G )
4 tkgeom.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( A I C ) )  ->  G  e. TarskiG )
6 tgbtwntriv2.1 . . . 4  |-  ( ph  ->  A  e.  P )
76adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( A I C ) )  ->  A  e.  P )
8 tgbtwntriv2.2 . . . 4  |-  ( ph  ->  B  e.  P )
98adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( A I C ) )  ->  B  e.  P )
10 tgbtwncomb.3 . . . 4  |-  ( ph  ->  C  e.  P )
1110adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( A I C ) )  ->  C  e.  P )
12 simpr 461 . . 3  |-  ( (
ph  /\  B  e.  ( A I C ) )  ->  B  e.  ( A I C ) )
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 24004 . 2  |-  ( (
ph  /\  B  e.  ( A I C ) )  ->  B  e.  ( C I A ) )
144adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  G  e. TarskiG )
1510adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  C  e.  P )
168adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  P )
176adantr 465 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  P )
18 simpr 461 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( C I A ) )
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 24004 . 2  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( A I C ) )
2013, 19impbida 832 1  |-  ( ph  ->  ( B  e.  ( A I C )  <-> 
B  e.  ( C I A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14643   distcds 14720  TarskiGcstrkg 23950  Itvcitv 23957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-trkgc 23969  df-trkgb 23970  df-trkgcb 23971  df-trkg 23975
This theorem is referenced by:  colcom  24070  colrot1  24071  lncom  24127  lnrot1  24128  lnrot2  24129  mirreu3  24160
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