Praeclarum Theorema

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Theorem

Formulation 1

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

Formulation 2

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

Also see

Compare the Constructive Dilemma, which is similar in appearance.

Historical Note

The Praeclarum Theorema was noted and named by Gottfried Wilhelm von Leibniz, who stated and proved it in the following manner:

If $a$ is $b$ and $d$ is $c$, then $ad$ will be $bc$.
This is a fine theorem, which is proved in this way:
$a$ is $b$, therefore $ad$ is $bd$ (by what precedes),
$d$ is $c$, therefore $bd$ is $bc$ (again by what precedes),
$ad$ is $bd$, and $bd$ is $bc$, therefore $ad$ is $bc$.
Q.E.D.

Linguistic Note

Praeclarum Theorema is Latin for splendid theorem.

It was so named by Gottfried Wilhelm von Leibniz.