Rule of Exportation

Formulation 2
This can of course be expressed as two separate theorems:

Reverse Implication
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 the tableau method of natural deduction:

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.

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

Also known as
This rule is also known as the rule of shunting.