Theorem tgcgrcoml

Description: Congruence commutes on the LHS. Variant of Theorem 2.5 of Schwabhauser p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020)

	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	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgcgrcomr.a	$$⊢\phi \to A\in P$$
	tgcgrcomr.b	$$⊢\phi \to B\in P$$
	tgcgrcomr.c	$$⊢\phi \to C\in P$$
	tgcgrcomr.d	$$⊢\phi \to D\in P$$
	tgcgrcomr.6	$$⊢\phi \to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
Assertion	tgcgrcoml	$$⊢\phi \to B\underset{\dot{}}{-}A=C\underset{\dot{}}{-}D$$

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		tgcgrcomr.a	$$⊢\phi \to A\in P$$
6		tgcgrcomr.b	$$⊢\phi \to B\in P$$
7		tgcgrcomr.c	$$⊢\phi \to C\in P$$
8		tgcgrcomr.d	$$⊢\phi \to D\in P$$
9		tgcgrcomr.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	10 9	eqtr3d	$$⊢\phi \to B\underset{\dot{}}{-}A=C\underset{\dot{}}{-}D$$