Theorem tgcgreqb

Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019)

	Ref	Expression
Hypotheses	tkgeom.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tkgeom.d	⊢ − = ( dist ‘ 𝐺 )
	tkgeom.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tkgeom.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgcgrcomlr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgcgrcomlr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgcgrcomlr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgcgrcomlr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgcgrcomlr.6	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
Assertion	tgcgreqb	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tkgeom.d	⊢ − = ( dist ‘ 𝐺 )
3		tkgeom.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tkgeom.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgcgrcomlr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgcgrcomlr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgcgrcomlr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgcgrcomlr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9		tgcgrcomlr.6	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
10	4	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐺 ∈ TarskiG )
11	7	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐶 ∈ 𝑃 )
12	8	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐷 ∈ 𝑃 )
13	6	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝑃 )
14	9	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
15		simpr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 )
16	15	oveq1d	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐵 ) )
17	14 16	eqtr3d	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝐶 − 𝐷 ) = ( 𝐵 − 𝐵 ) )
18	1 2 3 10 11 12 13 17	axtgcgrid	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐶 = 𝐷 )
19	4	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐺 ∈ TarskiG )
20	5	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐴 ∈ 𝑃 )
21	6	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐵 ∈ 𝑃 )
22	8	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐷 ∈ 𝑃 )
23	9	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
24		simpr	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐶 = 𝐷 )
25	24	oveq1d	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → ( 𝐶 − 𝐷 ) = ( 𝐷 − 𝐷 ) )
26	23 25	eqtrd	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐷 ) )
27	1 2 3 19 20 21 22 26	axtgcgrid	⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐴 = 𝐵 )
28	18 27	impbida	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) )