Theorem tgcgrcomimp

Description: Congruence commutes on the RHS. Theorem 2.5 of Schwabhauser p. 27. (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	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgcgrcomimp.a	$$⊢\phi \to A\in P$$
	tgcgrcomimp.b	$$⊢\phi \to B\in P$$
	tgcgrcomimp.c	$$⊢\phi \to C\in P$$
	tgcgrcomimp.d	$$⊢\phi \to D\in P$$
Assertion	tgcgrcomimp	$$⊢\phi \to \left(A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}C\right)$$

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		tgcgrcomimp.a	$$⊢\phi \to A\in P$$
6		tgcgrcomimp.b	$$⊢\phi \to B\in P$$
7		tgcgrcomimp.c	$$⊢\phi \to C\in P$$
8		tgcgrcomimp.d	$$⊢\phi \to D\in P$$
9	1 2 3 4 7 8	axtgcgrrflx	$$⊢\phi \to C\underset{\dot{}}{-}D=D\underset{\dot{}}{-}C$$
10	9	eqeq2d	$$⊢\phi \to \left(A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\leftrightarrow A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}C\right)$$
11	10	biimpd	$$⊢\phi \to \left(A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}C\right)$$