Proof by Cases/Formulation 1/Forward Implication/Proof 2

Theorem

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

Proof
From the Constructive Dilemma we have:
 * $p \implies q, r \implies s \vdash p \lor r \implies q \lor s$

from which, changing the names of letters strategically:
 * $p \implies r, q \implies r \vdash p \lor q \implies r \lor r$

From the Rule of Idempotence we have:
 * $r \lor r \vdash r$

and the result follows by Hypothetical Syllogism.