Theorem tgcgreqb

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	$$⊢\phi \to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
Assertion	tgcgreqb	$$⊢\phi \to \left(A=B\leftrightarrow C=D\right)$$

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		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	4	adantr	$$⊢\left(\phi \wedge A=B\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
11	7	adantr	$$⊢\left(\phi \wedge A=B\right)\to C\in P$$
12	8	adantr	$$⊢\left(\phi \wedge A=B\right)\to D\in P$$
13	6	adantr	$$⊢\left(\phi \wedge A=B\right)\to B\in P$$
14	9	adantr	$$⊢\left(\phi \wedge A=B\right)\to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
15		simpr	$$⊢\left(\phi \wedge A=B\right)\to A=B$$
16	15	oveq1d	$$⊢\left(\phi \wedge A=B\right)\to A\underset{\dot{}}{-}B=B\underset{\dot{}}{-}B$$
17	14 16	eqtr3d	$$⊢\left(\phi \wedge A=B\right)\to C\underset{\dot{}}{-}D=B\underset{\dot{}}{-}B$$
18	1 2 3 10 11 12 13 17	axtgcgrid	$$⊢\left(\phi \wedge A=B\right)\to C=D$$
19	4	adantr	$$⊢\left(\phi \wedge C=D\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
20	5	adantr	$$⊢\left(\phi \wedge C=D\right)\to A\in P$$
21	6	adantr	$$⊢\left(\phi \wedge C=D\right)\to B\in P$$
22	8	adantr	$$⊢\left(\phi \wedge C=D\right)\to D\in P$$
23	9	adantr	$$⊢\left(\phi \wedge C=D\right)\to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
24		simpr	$$⊢\left(\phi \wedge C=D\right)\to C=D$$
25	24	oveq1d	$$⊢\left(\phi \wedge C=D\right)\to C\underset{\dot{}}{-}D=D\underset{\dot{}}{-}D$$
26	23 25	eqtrd	$$⊢\left(\phi \wedge C=D\right)\to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}D$$
27	1 2 3 19 20 21 22 26	axtgcgrid	$$⊢\left(\phi \wedge C=D\right)\to A=B$$
28	18 27	impbida	$$⊢\phi \to \left(A=B\leftrightarrow C=D\right)$$