Exportation^[1][2][3][4] is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs. It is the rule that:

${\displaystyle ((P\land Q)\to R)\Leftrightarrow (P\to (Q\to R))}$

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Exportation

Type	Rule of replacement
Field	Propositional calculus
Symbolic statement	${\displaystyle ((P\land Q)\to R)\Leftrightarrow (P\to (Q\to R))}$

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Where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, ${\displaystyle ((P\land Q)\to R)\Rightarrow (P\to (Q\to R))}$ is the law of exportation, for it "exports" a proposition from the antecedent of ${\displaystyle (P\land Q)\to R}$ to its consequent. Its converse, the law of importation, ${\displaystyle (P\to (Q\to R))\Rightarrow ((P\land Q)\to R)}$, "imports" a proposition from the consequent of ${\displaystyle P\to (Q\to R)}$ to its antecedent.

The exportation rule may be written in sequent notation:

${\displaystyle ((P\land Q)\to R)\dashv \vdash (P\to (Q\to R))}$

where ${\displaystyle \dashv \vdash }$ is a metalogical symbol meaning that ${\displaystyle (P\to (Q\to R))}$ is a syntactic equivalent of ${\displaystyle ((P\land Q)\to R)}$ in some logical system;

or in rule form:

${\displaystyle {\frac {(P\land Q)\to R}{P\to (Q\to R)}}}$, ${\displaystyle {\frac {P\to (Q\to R)}{(P\land Q)\to R}}.}$

where the rule is that wherever an instance of "${\displaystyle (P\land Q)\to R}$" appears on a line of a proof, it can be replaced with "${\displaystyle P\to (Q\to R)}$", and vice versa.

Import-export is a name given to the statement as a theorem or truth-functional tautology of propositional logic:

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

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

The following proof uses a classically valid chain of equivalences. Rules used are material implication, De Morgan's law, and the associative property of conjunction.

More information , ...

Proposition	Derivation
${\displaystyle P\rightarrow (Q\rightarrow R)}$	Given
${\displaystyle \neg P\lor (Q\rightarrow R)}$	material implication
${\displaystyle \neg P\lor (\neg Q\lor R)}$	material implication
${\displaystyle (\neg P\lor \neg Q)\lor R}$	associativity
${\displaystyle \neg (P\land Q)\lor R}$	De Morgan's law
${\displaystyle (P\land Q)\rightarrow R}$	material implication

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Exportation is associated with currying via the Curry–Howard correspondence.^{[citation needed]}