Proof by Cases/Formulation 1/Forward Implication/Proof 2
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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.
$\blacksquare$