Conversion per Accidens

Theorem
Consider the categorical statements:
 * $\mathbf A \left({S, P}\right): \quad$ The universal affirmative: $\forall x: S \left({x}\right) \implies P \left({x}\right)$
 * $\mathbf I \left({P, S}\right): \quad$ The particular affirmative: $\exists x: P \left({x}\right) \land S \left({x}\right)$

Then:
 * $\mathbf A \left({S, P}\right) \implies \mathbf I \left({P, S}\right)$

iff:
 * $\exists x: S \left({x}\right)$

Using the symbology of predicate logic:
 * $\exists x: S \left({x}\right) \iff \left({\left({\forall x: S \left({x}\right) \implies P \left({x}\right)}\right) \implies \left({\exists x: P \left({x}\right) \land S \left({x}\right)}\right)}\right)$

This law has the traditional name conversion per accidens of $\mathbf A$.

Thus the $\mathbf A$ form converts per accidens to the $\mathbb I$ form.

Proof
From Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous:
 * $\exists x: S \left({x}\right) \iff \left({\left({\forall x: S \left({x}\right) \implies P \left({x}\right)}\right) \implies \left({\exists x: S \left({x}\right) \land P \left({x}\right)}\right)}\right)$

From Law of Simple Conversion of I:
 * $\left({\exists x: S \left({x}\right) \land P \left({x}\right)}\right) \implies \left({\exists x: P \left({x}\right) \land S \left({x}\right)}\right)$

Hence the result.

Also defined as
Some sources gloss over the possibility of $S \left({x}\right)$ being vacuous and merely report that the universal affirmative $\mathbf A \left({S, P}\right)$ implies the particular affirmative $\mathbf I \left({P, S}\right)$.