Biconditional introduction

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

In propositional logic, biconditional introduction is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements. The rule makes it possible to introduce a biconditional statement into a logical proof. If $P\to Q$ is true, and if $Q\to P$ is true, then one may infer that $P\leftrightarrow Q$ is true. For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive". Biconditional introduction is the converse of biconditional elimination. The rule can be stated formally as:

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

where the rule is that wherever instances of "$P\to Q$" and "$Q\to P$" appear on lines of a proof, "$P\leftrightarrow Q$" can validly be placed on a subsequent line.

Formal notation

The biconditional introduction rule may be written in sequent notation:

$(P\to Q),(Q\to P)\vdash (P\leftrightarrow Q)$

where $\vdash$ is a metalogical symbol meaning that $P\leftrightarrow Q$ is a syntactic consequence when $P\to Q$ and $Q\to P$ are both in a proof;

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

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

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