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

Theorem

$\paren {p \implies r} \land \paren {q \implies r} \vdash \paren {p \lor q} \implies r$

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.

$\blacksquare$