Biconditional elimination

Transformation rules
Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

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.

See also

• Logical biconditional