Rule of Exportation

Theorem

 * $\left ({p \land q}\right) \implies r \dashv \vdash p \implies \left ({q \implies r}\right)$

Also known as the rule of shunting.

Alternative rendition
It can alternatively be rendered as:


 * $\vdash \left({\left ({p \land q}\right) \implies r}\right) \iff \left({p \implies \left ({q \implies r}\right)}\right)$

It can be seen to be logically equivalent to the form above.

Proof by Natural Deduction
By the tableau method:

Proof by Truth Table
We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connectives match for all models.

$\begin{array}{|ccccc||ccccc|} \hline (p & \land & q) & \implies & r & p & \implies & (q & \implies & r) \\ \hline F & F & F & T & F & F & T & F & T & F \\ F & F & F & T & T & F & T & F & T & T \\ F & F & T & T & F & F & T & T & F & F \\ F & F & T & T & T & F & T & T & T & T \\ T & F & F & T & F & T & T & F & T & F \\ T & F & F & T & T & T & T & F & T & T \\ T & T & T & F & F & T & F & T & F & F \\ T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$