Theorem tgbtwncomb 25384

Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgbtwntriv2.1	|- ( ph -> A e. P )
tgbtwntriv2.2	|- ( ph -> B e. P )
tgbtwncomb.3	|- ( ph -> C e. P )

Assertion

Ref	Expression
tgbtwncomb	|- ( ph -> ( B e. ( A I C ) <-> B e. ( C I A ) ) )

Proof of Theorem tgbtwncomb

Step	Hyp	Ref	Expression
1		tkgeom.p	. . 3 |- P = ( Base ` G )
2		tkgeom.d	. . 3 |- .- = ( dist ` G )
3		tkgeom.i	. . 3 |- I = (Itv ` G )
4		tkgeom.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . 3 |- ( ( ph /\ B e. ( A I C ) ) -> G e. TarskiG )
6		tgbtwntriv2.1	. . . 4 |- ( ph -> A e. P )
7	6	adantr 481	. . 3 |- ( ( ph /\ B e. ( A I C ) ) -> A e. P )
8		tgbtwntriv2.2	. . . 4 |- ( ph -> B e. P )
9	8	adantr 481	. . 3 |- ( ( ph /\ B e. ( A I C ) ) -> B e. P )
10		tgbtwncomb.3	. . . 4 |- ( ph -> C e. P )
11	10	adantr 481	. . 3 |- ( ( ph /\ B e. ( A I C ) ) -> C e. P )
12		simpr 477	. . 3 |- ( ( ph /\ B e. ( A I C ) ) -> B e. ( A I C ) )
13	1, 2, 3, 5, 7, 9, 11, 12	tgbtwncom 25383	. 2 |- ( ( ph /\ B e. ( A I C ) ) -> B e. ( C I A ) )
14	4	adantr 481	. . 3 |- ( ( ph /\ B e. ( C I A ) ) -> G e. TarskiG )
15	10	adantr 481	. . 3 |- ( ( ph /\ B e. ( C I A ) ) -> C e. P )
16	8	adantr 481	. . 3 |- ( ( ph /\ B e. ( C I A ) ) -> B e. P )
17	6	adantr 481	. . 3 |- ( ( ph /\ B e. ( C I A ) ) -> A e. P )
18		simpr 477	. . 3 |- ( ( ph /\ B e. ( C I A ) ) -> B e. ( C I A ) )
19	1, 2, 3, 14, 15, 16, 17, 18	tgbtwncom 25383	. 2 |- ( ( ph /\ B e. ( C I A ) ) -> B e. ( A I C ) )
20	13, 19	impbida 877	1 |- ( ph -> ( B e. ( A I C ) <-> B e. ( C I A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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:  colcom  25453  colrot1  25454  lnhl  25510  lncom  25517  lnrot1  25518  lnrot2  25519  mirreu3  25549