Exportation (logic)
Exportation 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:

Where "" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology,  is the law of exportation, for it "exports" a proposition from the antecedent of  to its consequent. Its converse, the law of importation, , "imports" a proposition from the consequent of  to its antecedent.

Formal notation 
The exportation rule may be written in sequent notation:

where  is a metalogical symbol meaning that  is a syntactic equivalent of  in some logical system;

or in rule form:
, 

where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "", and vice versa.

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

where , , and  are propositions expressed in some logical system.

Natural language

Truth values 
At any time, if P→Q is true, it can be replaced by P→(P∧Q). 
One possible case for P→Q is for P to be true and Q to be true; thus P∧Q is also true, and P→(P∧Q) is true.  
Another possible case sets P as false and Q as true. Thus, P∧Q is false and P→(P∧Q) is false; false→false is true. 
The last case occurs when both P and Q are false. Thus, P∧Q is false and P→(P∧Q) is true.

Example 
It rains and the sun shines implies that there is a rainbow. 
Thus, if it rains, then the sun shines implies that there is a rainbow.

If my car is on, when I switch the gear to D the car starts going.
If my car is on and I have switched the gear to D, then the car must start going.

Proof  

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.

Relation to functions
Exportation is associated with currying via the Curry–Howard correspondence.

References

Rules of inference
Theorems in propositional logic