Theorem cgr3id

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

	Ref	Expression
Hypotheses	tgcgrxfr.p	$$⊢P={\mathrm{Base}}_G$$
	tgcgrxfr.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgcgrxfr.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgcgrxfr.r	$$⊢\underset{\dot{}}{\sim }={\sim }_{𝒢}\left(G\right)$$
	tgcgrxfr.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgbtwnxfr.a	$$⊢\phi \to A\in P$$
	tgbtwnxfr.b	$$⊢\phi \to B\in P$$
	tgbtwnxfr.c	$$⊢\phi \to C\in P$$
Assertion	cgr3id	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“ABC”⟩$$

Proof

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	$$⊢P={\mathrm{Base}}_G$$
2		tgcgrxfr.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgcgrxfr.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgcgrxfr.r	$$⊢\underset{\dot{}}{\sim }={\sim }_{𝒢}\left(G\right)$$
5		tgcgrxfr.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		tgbtwnxfr.a	$$⊢\phi \to A\in P$$
7		tgbtwnxfr.b	$$⊢\phi \to B\in P$$
8		tgbtwnxfr.c	$$⊢\phi \to C\in P$$
9		eqidd	$$⊢\phi \to A\underset{\dot{}}{-}B=A\underset{\dot{}}{-}B$$
10		eqidd	$$⊢\phi \to B\underset{\dot{}}{-}C=B\underset{\dot{}}{-}C$$
11		eqidd	$$⊢\phi \to C\underset{\dot{}}{-}A=C\underset{\dot{}}{-}A$$
12	1 2 4 5 6 7 8 6 7 8 9 10 11	trgcgr	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“ABC”⟩$$