Metamath Proof Explorer

Theorem btwnswapid2

Description: If you can swap arguments one and three of a betweenness statement, then those arguments are identical. (Contributed by Scott Fenton, 7-Oct-2013)

Ref Expression
Assertion btwnswapid2 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ∧ C Btwn B A → A = C

Proof

Step Hyp Ref Expression
1 btwncom ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ↔ A Btwn C B
2 3anrev ⊢ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ↔ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N
3 btwncom ⊢ N ∈ ℕ ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N → C Btwn B A ↔ C Btwn A B
4 2 3 sylan2b ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn B A ↔ C Btwn A B
5 1 4 anbi12d ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ∧ C Btwn B A ↔ A Btwn C B ∧ C Btwn A B
6 3ancomb ⊢ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ↔ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N
7 btwnswapid ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N → A Btwn C B ∧ C Btwn A B → A = C
8 6 7 sylan2b ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn C B ∧ C Btwn A B → A = C
9 5 8 sylbid ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ∧ C Btwn B A → A = C