Theorem cgr3id

Description: Reflexivity law for three-place congruence. (Contributed by Thierry Arnoux, 28-Apr-2019)

	Ref	Expression
Hypotheses	tgcgrxfr.p	|- P = ( Base ` G )
	tgcgrxfr.m	|- .- = ( dist ` G )
	tgcgrxfr.i	|- I = ( Itv ` G )
	tgcgrxfr.r	|- .~ = ( cgrG ` G )
	tgcgrxfr.g	|- ( ph -> G e. TarskiG )
	tgbtwnxfr.a	|- ( ph -> A e. P )
	tgbtwnxfr.b	|- ( ph -> B e. P )
	tgbtwnxfr.c	|- ( ph -> C e. P )
Assertion	cgr3id	|- ( ph -> <" A B C "> .~ <" A B C "> )

Proof

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	|- P = ( Base ` G )
2		tgcgrxfr.m	|- .- = ( dist ` G )
3		tgcgrxfr.i	|- I = ( Itv ` G )
4		tgcgrxfr.r	|- .~ = ( cgrG ` G )
5		tgcgrxfr.g	|- ( ph -> G e. TarskiG )
6		tgbtwnxfr.a	|- ( ph -> A e. P )
7		tgbtwnxfr.b	|- ( ph -> B e. P )
8		tgbtwnxfr.c	|- ( ph -> C e. P )
9		eqidd	|- ( ph -> ( A .- B ) = ( A .- B ) )
10		eqidd	|- ( ph -> ( B .- C ) = ( B .- C ) )
11		eqidd	|- ( ph -> ( C .- A ) = ( C .- A ) )
12	1 2 4 5 6 7 8 6 7 8 9 10 11	trgcgr	|- ( ph -> <" A B C "> .~ <" A B C "> )