Self-Distributive Law for Conditional

Theorem
The following is known as the Self-Distributive Law:


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

We also have, interestingly, this result:


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

... but:


 * $$\left({p \implies r}\right) \implies \left({q \implies r}\right) \not \vdash \left({p \implies q}\right) \implies r$$

Proof by Natural Deduction
These are proved by the Tableau method.

Proof by Truth Table
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.

Now we investigate the relative truths of $$\left({p \implies q}\right) \implies r$$ and $$\left({p \implies r}\right) \implies \left({q \implies r}\right)$$.

As can be seen, the final two columns do not match. Therefore the two expressions are not logically equivalent.

However, note that there are no instances where $$v \left({\left({p \implies q}\right) \implies r}\right) = T$$ at the same time that $$v \left({\left({p \implies r}\right) \implies \left({q \implies r}\right)}\right)$$.

Hence (indirectly) the result:
 * $$\left({p \implies q}\right) \implies r \vdash \left({p \implies r}\right) \implies \left({q \implies r}\right)$$