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Theorem tgbtwncomb 22942
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 22941 . 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 22941 . 2  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( A I C ) )
2013, 19impbida 828 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 1369    e. wcel 1756   ` cfv 5417  (class class class)co 6090   Basecbs 14173   distcds 14246  TarskiGcstrkg 22888  Itvcitv 22896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4420
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093  df-trkgc 22908  df-trkgb 22909  df-trkgcb 22910  df-trkg 22915
This theorem is referenced by:  colcom  22991  colrot1  22992  lncom  23028  lnrot1  23029  lnrot2  23030  mirreu3  23057
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