Theorem tgbtwnexch3 23075

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 23069	. 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 23069	. 2 |- ( ph -> C e. ( D I A ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 23074	1 |- ( ph -> C e. ( B I D ) )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   distcds 14358  TarskiGcstrkg 23015  Itvcitv 23022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-trkgc 23034  df-trkgb 23035  df-trkgcb 23036  df-trkg 23040

This theorem is referenced by:  tgbtwnouttr2  23076  tgbtwnexch2  23077  tgifscgr  23090  tgcgrxfr  23099  tgbtwnconn1lem1  23134  tgbtwnconn1lem2  23135  tgbtwnconn1lem3  23136  tgbtwnconn2  23138  tgbtwnconn3  23139  tglineeltr  23169  miriso  23209  krippenlem  23220