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Theorem tgbtwnexch3 24010
Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p  |-  P  =  ( Base `  G
)
tkgeom.d  |-  .-  =  ( dist `  G )
tkgeom.i  |-  I  =  (Itv `  G )
tkgeom.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwnintr.1  |-  ( ph  ->  A  e.  P )
tgbtwnintr.2  |-  ( ph  ->  B  e.  P )
tgbtwnintr.3  |-  ( ph  ->  C  e.  P )
tgbtwnintr.4  |-  ( ph  ->  D  e.  P )
tgbtwnexch3.5  |-  ( ph  ->  B  e.  ( A I C ) )
tgbtwnexch3.6  |-  ( ph  ->  C  e.  ( A I D ) )
Assertion
Ref Expression
tgbtwnexch3  |-  ( ph  ->  C  e.  ( B I D ) )

Proof of Theorem tgbtwnexch3
StepHypRef Expression
1 tkgeom.p . 2  |-  P  =  ( Base `  G
)
2 tkgeom.d . 2  |-  .-  =  ( dist `  G )
3 tkgeom.i . 2  |-  I  =  (Itv `  G )
4 tkgeom.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 tgbtwnintr.2 . 2  |-  ( ph  ->  B  e.  P )
6 tgbtwnintr.3 . 2  |-  ( ph  ->  C  e.  P )
7 tgbtwnintr.4 . 2  |-  ( ph  ->  D  e.  P )
8 tgbtwnintr.1 . 2  |-  ( ph  ->  A  e.  P )
9 tgbtwnexch3.5 . . 3  |-  ( ph  ->  B  e.  ( A I C ) )
101, 2, 3, 4, 8, 5, 6, 9tgbtwncom 24004 . 2  |-  ( ph  ->  B  e.  ( C I A ) )
11 tgbtwnexch3.6 . . 3  |-  ( ph  ->  C  e.  ( A I D ) )
121, 2, 3, 4, 8, 6, 7, 11tgbtwncom 24004 . 2  |-  ( ph  ->  C  e.  ( D I A ) )
131, 2, 3, 4, 5, 6, 7, 8, 10, 12tgbtwnintr 24009 1  |-  ( ph  ->  C  e.  ( B I D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = 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:  tgbtwnouttr2  24011  tgifscgr  24025  tgcgrxfr  24034  tgbtwnconn1lem1  24084  tgbtwnconn1lem2  24085  tgbtwnconn1lem3  24086  tgbtwnconn2  24088  tgbtwnconn3  24089  btwnhl  24123  tglineeltr  24136  miriso  24175  krippenlem  24192
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