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 $$P \leftrightarrow Q$$ is true, then one may infer that $$P \to Q$$ is true, and also that $$Q \to P$$ 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:


 * $$\frac{P \leftrightarrow Q}{\therefore P \to Q}$$

and
 * $$\frac{P \leftrightarrow Q}{\therefore Q \to P}$$

where the rule is that wherever an instance of "$$P \leftrightarrow Q$$" appears on a line of a proof, either "$$P \to Q$$" or "$$Q \to P$$" can be placed on a subsequent line.

Formal notation
The biconditional elimination rule may be written in sequent notation:
 * $$(P \leftrightarrow Q) \vdash (P \to Q)$$

and
 * $$(P \leftrightarrow Q) \vdash (Q \to P)$$

where $$\vdash$$ is a metalogical symbol meaning that $$P \to Q$$, in the first case, and $$Q \to P$$ in the other are syntactic consequences of $$P \leftrightarrow Q$$ in some logical system;

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


 * $$(P \leftrightarrow Q) \to (P \to Q)$$
 * $$(P \leftrightarrow Q) \to (Q \to P)$$

where $$P$$, and $$Q$$ are propositions expressed in some formal system.