Theorem tgbtwnexch2 25391

Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-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 )
tgbtwnexch2.1	|- ( ph -> B e. ( A I D ) )
tgbtwnexch2.2	|- ( ph -> C e. ( B I D ) )

Assertion

Ref	Expression
tgbtwnexch2	|- ( ph -> C e. ( A I D ) )

Proof of Theorem tgbtwnexch2

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ph /\ B = C ) -> B = C )
2		tgbtwnexch2.1	. . . 4 |- ( ph -> B e. ( A I D ) )
3	2	adantr 481	. . 3 |- ( ( ph /\ B = C ) -> B e. ( A I D ) )
4	1, 3	eqeltrrd 2702	. 2 |- ( ( ph /\ B = C ) -> C e. ( A I D ) )
5		tkgeom.p	. . 3 |- P = ( Base ` G )
6		tkgeom.d	. . 3 |- .- = ( dist ` G )
7		tkgeom.i	. . 3 |- I = (Itv ` G )
8		tkgeom.g	. . . 4 |- ( ph -> G e. TarskiG )
9	8	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> G e. TarskiG )
10		tgbtwnintr.1	. . . 4 |- ( ph -> A e. P )
11	10	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> A e. P )
12		tgbtwnintr.2	. . . 4 |- ( ph -> B e. P )
13	12	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> B e. P )
14		tgbtwnintr.3	. . . 4 |- ( ph -> C e. P )
15	14	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> C e. P )
16		tgbtwnintr.4	. . . 4 |- ( ph -> D e. P )
17	16	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> D e. P )
18		simpr 477	. . 3 |- ( ( ph /\ B =/= C ) -> B =/= C )
19		tgbtwnexch2.2	. . . . . 6 |- ( ph -> C e. ( B I D ) )
20	19	adantr 481	. . . . 5 |- ( ( ph /\ B =/= C ) -> C e. ( B I D ) )
21	2	adantr 481	. . . . 5 |- ( ( ph /\ B =/= C ) -> B e. ( A I D ) )
22	5, 6, 7, 9, 15, 13, 11, 17, 20, 21	tgbtwnintr 25388	. . . 4 |- ( ( ph /\ B =/= C ) -> B e. ( C I A ) )
23	5, 6, 7, 9, 15, 13, 11, 22	tgbtwncom 25383	. . 3 |- ( ( ph /\ B =/= C ) -> B e. ( A I C ) )
24	5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20	tgbtwnouttr2 25390	. 2 |- ( ( ph /\ B =/= C ) -> C e. ( A I D ) )
25	4, 24	pm2.61dane 2881	1 |- ( ph -> C e. ( A I D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` 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-ne 2795  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:  tgbtwnexch  25393  tgtrisegint  25394  tgbtwnconn1lem3  25469  legtri3  25485  miriso  25565