Theorem tgbtwnexch3 23613

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 23607	. 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 23607	. 2 |- ( ph -> C e. ( D I A ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 23612	1 |- ( ph -> C e. ( B I D ) )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   distcds 14560  TarskiGcstrkg 23553  Itvcitv 23560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-trkgc 23572  df-trkgb 23573  df-trkgcb 23574  df-trkg 23578

This theorem is referenced by:  tgbtwnouttr2  23614  tgbtwnexch2  23615  tgifscgr  23628  tgcgrxfr  23637  tgbtwnconn1lem1  23686  tgbtwnconn1lem2  23687  tgbtwnconn1lem3  23688  tgbtwnconn2  23690  tgbtwnconn3  23691  tglineeltr  23725  miriso  23763  krippenlem  23775