Self-Distributive Law for Conditional/Forward Implication/Formulation 2/Proof by Truth Table

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the truth value under the main connective is true for all boolean interpretations.

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