Theorem tgcgrcomlr

Description: Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	$$⊢P={\mathrm{Base}}_G$$
	tkgeom.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tkgeom.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tkgeom.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgcgrcomlr.a	$$⊢\phi \to A\in P$$
	tgcgrcomlr.b	$$⊢\phi \to B\in P$$
	tgcgrcomlr.c	$$⊢\phi \to C\in P$$
	tgcgrcomlr.d	$$⊢\phi \to D\in P$$
	tgcgrcomlr.6	$$⊢\phi \to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
Assertion	tgcgrcomlr	$$⊢\phi \to B\underset{\dot{}}{-}A=D\underset{\dot{}}{-}C$$

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	$$⊢P={\mathrm{Base}}_G$$
2		tkgeom.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tkgeom.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tkgeom.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgcgrcomlr.a	$$⊢\phi \to A\in P$$
6		tgcgrcomlr.b	$$⊢\phi \to B\in P$$
7		tgcgrcomlr.c	$$⊢\phi \to C\in P$$
8		tgcgrcomlr.d	$$⊢\phi \to D\in P$$
9		tgcgrcomlr.6	$$⊢\phi \to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
10	1 2 3 4 5 6	axtgcgrrflx	$$⊢\phi \to A\underset{\dot{}}{-}B=B\underset{\dot{}}{-}A$$
11	1 2 3 4 7 8	axtgcgrrflx	$$⊢\phi \to C\underset{\dot{}}{-}D=D\underset{\dot{}}{-}C$$
12	9 10 11	3eqtr3d	$$⊢\phi \to B\underset{\dot{}}{-}A=D\underset{\dot{}}{-}C$$