Praeclarum Theorema

Theorem

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

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

Compare the Constructive Dilemma, which is similar in appearance.

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.

Praeclarum Theorema is Latin for splendid theorem.