Metamath Proof Explorer

Theorem btwncom

Description: Betweenness commutes. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion btwncom ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ↔ A Btwn C B

Proof

Step Hyp Ref Expression
1 btwncomim ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C → A Btwn C B
2 3ancomb ⊢ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ↔ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N
3 btwncomim ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N → A Btwn C B → A Btwn B C
4 2 3 sylan2b ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn C B → A Btwn B C
5 1 4 impbid ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ↔ A Btwn C B