Theorem tgbtwnexch3 25389

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 25383	. 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 25383	. 2 |- ( ph -> C e. ( D I A ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 25388	1 |- ( ph -> C e. ( B I D ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tgbtwnouttr2  25390  tgifscgr  25403  tgcgrxfr  25413  tgbtwnconn1lem1  25467  tgbtwnconn1lem2  25468  tgbtwnconn1lem3  25469  tgbtwnconn2  25471  tgbtwnconn3  25472  btwnhl  25509  tglineeltr  25526  miriso  25565  krippenlem  25585  outpasch  25647  hlpasch  25648