# 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.