Hypothetical Syllogism/Formulation 2/Proof by Truth Table

Theorem

 * $p \implies q, q \implies r, p \vdash r$

Proof
We apply the Method of Truth Tables to the propositions in turn.

As can be seen for all models by inspection, where the truth values under the main connective on the LHS (that is, the rightmost $\land$) is $T$, that under the instance of $r$ on the RHS is also $T$:

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

Hence the result.