section_path: Methods of proof/Closed chain inference
section_name: Closed chain inference
paragraph_text: Closed chain inference
  
  Main article: Closed chain inference
  
  A closed chain inference shows that a collection of statements are pairwise equivalent.
  
  In order to prove that the statements φ 1, …, φ n {\displaystyle \varphi _{1},\ldots,\varphi _{n}} are each pairwise equivalent, proofs are given for the implications φ 1 ⇒ φ 2 {\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}, φ 2 ⇒ φ 3 {\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}, … {\displaystyle \dots }, φ n − 1 ⇒ φ n {\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}} and φ n ⇒ φ 1 {\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}.
  
  The pairwise equivalence of the statements then results from the transitivity of the material conditional.