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
Let $$v: \left\{{p, q, r}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula of three variables $$p, q, r$$.

We see that $$v \left({\left ({p \and q}\right) \implies r}\right) = v \left({p \implies \left ({q \implies r}\right)}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.