Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous

Theorem
Consider the categorical statements:
 * $\mathbf I: \quad$ The particular affirmative: $\exists x: S \left({x}\right) \land P \left({x}\right)$
 * $\mathbf O: \quad$ The particular negative: $\exists x: S \left({x}\right) \land \neg P \left({x}\right)$

Then $\mathbf I$ and $\mathbf O$ are subcontrary iff $\exists x: S \left({x}\right)$.

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

Sufficient Condition
Let $\exists x: S \left({x}\right)$.

Suppose $\mathbf I$ and $\mathbf O$ are both false.

As $\mathbf I$ is false, then by the Rule of Conjunction:
 * $\neg P \left({x}\right)$

As $\mathbf O$ is true, then by the Rule of Conjunction:
 * $\neg \neg P \left({x}\right)$

and so by Double Negation:
 * $P \left({x}\right)$

It follows by Proof by Contradiction that $\mathbf I$ and $\mathbf O$ are not both false.

Thus, by definition, $\mathbf I$ and $\mathbf O$ are subcontrary statements.

Necessary Condition
Let $\mathbf I$ and $\mathbf O$ be subcontrary statements

Suppose $\neg \exists x: S \left({x}\right)$.

From the definition of logical conjunction, it follows that:
 * $\neg \left({\exists x: S \left({x}\right) \land P \left({x}\right)}\right)$

Similarly from the definition of logical conjunction, it follows that:
 * $\neg \left({\exists x: S \left({x}\right) \land \neg P \left({x}\right)}\right)$

That is, both $\mathbf I$ and $\mathbf O$ are false.

So, by definition, $\mathbf I$ and $\mathbf O$ are not subcontrary.

It follows by Proof by Contradiction that $\exists x: S \left({x}\right)$.

Also defined as
Some sources gloss over the possibility of $S \left({x}\right)$ being vacuous and merely report that the particular affirmative and particular negative are subcontrary.