Biconditional elimination

Biconditional elimination is the name of two valid rules of inference of propositional logic. It allows for one to infer a conditional from a biconditional. If is true, then one may infer that is true, and also that is true. For example, if it's true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, I'm alive; likewise, it's true that if I'm alive, I'm breathing. The rules can be stated formally as:


where the rule is that wherever an instance of "" appears on a line of a proof, either "" or "" can be placed on a subsequent line;

Formal notation

The biconditional elimination rule may be written in sequent notation:


where is a metalogical symbol meaning that , in the first case, and in the other are syntactic consequences of in some logical system;

or as the statement of a truth-functional tautology or theorem of propositional logic:

where , and are propositions expressed in some formal system.

See also