Hypothetical Syllogism

Definition

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

Its abbreviation in a tableau proof is $$\textrm{HS}$$.

It is otherwise known as the transitivity law.

Proof by Natural Deduction
By the tableau method:

Proof by Truth Table
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 connectives on the LHS is $$T$$, that under the one on the RHS is also $$T$$:

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

Hence the result.