Theorem tgcgreq

Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-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	$$⊢\phi \to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
	tgcgreq.1	$$⊢\phi \to A=B$$
Assertion	tgcgreq	$$⊢\phi \to C=D$$

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		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		tgcgreq.1	$$⊢\phi \to A=B$$
11	1 2 3 4 5 6 7 8 9	tgcgreqb	$$⊢\phi \to \left(A=B\leftrightarrow C=D\right)$$
12	10 11	mpbid	$$⊢\phi \to C=D$$