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→ABtwnBC∧CBtwnBA→A=C

Proof

Step Hyp Ref Expression
1 btwncom ⊢N∈ℕ∧A∈𝔼⁡N∧B∈𝔼⁡N∧C∈𝔼⁡N→ABtwnBC↔ABtwnCB
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→CBtwnBA↔CBtwnAB
4 2 3 sylan2b ⊢N∈ℕ∧A∈𝔼⁡N∧B∈𝔼⁡N∧C∈𝔼⁡N→CBtwnBA↔CBtwnAB
5 1 4 anbi12d ⊢N∈ℕ∧A∈𝔼⁡N∧B∈𝔼⁡N∧C∈𝔼⁡N→ABtwnBC∧CBtwnBA↔ABtwnCB∧CBtwnAB
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→ABtwnCB∧CBtwnAB→A=C
8 6 7 sylan2b ⊢N∈ℕ∧A∈𝔼⁡N∧B∈𝔼⁡N∧C∈𝔼⁡N→ABtwnCB∧CBtwnAB→A=C
9 5 8 sylbid ⊢N∈ℕ∧A∈𝔼⁡N∧B∈𝔼⁡N∧C∈𝔼⁡N→ABtwnBC∧CBtwnBA→A=C