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

Step	Hyp	Ref	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 ) )
10	1, 2, 3, 4, 8, 5, 6, 9	tgbtwncom 24004	. 2 |- ( ph -> B e. ( C I A ) )
11		tgbtwnexch3.6	. . 3 |- ( ph -> C e. ( A I D ) )
12	1, 2, 3, 4, 8, 6, 7, 11	tgbtwncom 24004	. 2 |- ( ph -> C e. ( D I A ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 24009	1 |- ( ph -> C e. ( B I D ) )

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