Rule of Exportation

Definition

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

Proof by Natural Deduction
By the tableau method:


 * $$\left ({p \and q}\right) \implies r \vdash p \implies \left ({q \implies r}\right)$$:


 * $$p \implies \left ({q \implies r}\right) \vdash \left ({p \and q}\right) \implies r $$:

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.

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