Conversion per Accidens

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

Then:
 * $\map {\mathbf A} {S, P} \implies \map {\mathbf I} {P, S}$


 * $\exists x: \map S x$
 * $\exists x: \map S x$

Using the symbology of predicate logic:
 * $\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \map P x} \implies \paren {\exists x: \map P x \land \map S x} }$

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

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

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

From Law of Simple Conversion of I:
 * $\paren {\exists x: \map S x \land \map P x} \implies \paren {\exists x: \map P x \land \map S x}$

Hence the result.

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