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 |
|
tgbtwntriv2.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
tgbtwntriv2.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
tgbtwncomb.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
8 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐺 ∈ TarskiG ) |
9 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐴 ∈ 𝑃 ) |
10 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ 𝑃 ) |
11 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐶 ∈ 𝑃 ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) |
13 |
1 2 3 8 9 10 11 12
|
tgbtwncom |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) |
14 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐺 ∈ TarskiG ) |
15 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐶 ∈ 𝑃 ) |
16 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐵 ∈ 𝑃 ) |
17 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐴 ∈ 𝑃 ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) |
19 |
1 2 3 14 15 16 17 18
|
tgbtwncom |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) |
20 |
13 19
|
impbida |
⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ↔ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) |