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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwnxfr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwnxfr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwnxfr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
Assertion	cgr3id	⊢ ( 𝜑 → ⟨“ 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		tgbtwnxfr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		tgbtwnxfr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
8		tgbtwnxfr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
9		eqidd	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐵 ) )
10		eqidd	⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐵 − 𝐶 ) )
11		eqidd	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐶 − 𝐴 ) )
12	1 2 4 5 6 7 8 6 7 8 9 10 11	trgcgr	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )