Biconditional elimination

Biconditional elimination

Type	Rule of inference
Field	Propositional calculus
Statement	If ${\displaystyle P\leftrightarrow Q}$ is true, then one may infer that ${\displaystyle P\to Q}$ is true, and also that ${\displaystyle Q\to P}$ is true.
Symbolic statement	${\displaystyle {\frac {P\leftrightarrow Q}{\therefore P\to Q}}}$ ${\displaystyle {\frac {P\leftrightarrow Q}{\therefore Q\to P}}}$

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 ${\displaystyle P\leftrightarrow Q}$ is true, then one may infer that ${\displaystyle P\to Q}$ is true, and also that ${\displaystyle Q\to P}$ is true.¹ 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:

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

and

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

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

The biconditional elimination rule may be written in sequent notation:

${\displaystyle (P\leftrightarrow Q)\vdash (P\to Q)}$

and

${\displaystyle (P\leftrightarrow Q)\vdash (Q\to P)}$

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

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

${\displaystyle (P\leftrightarrow Q)\to (P\to Q)}$

${\displaystyle (P\leftrightarrow Q)\to (Q\to P)}$

where ${\displaystyle P}$, and ${\displaystyle Q}$ are propositions expressed in some formal system.