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

Theorem
Consider the categorical statements:

Then:
 * $\mathbf I$ and $\mathbf O$ are subcontrary


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

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

Sufficient Condition
Let $\exists x: \map S x$.

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

As $\mathbf I$ is false, then by the Rule of Conjunction:
 * $\neg \map P x$

As $\mathbf O$ is false, then by the Rule of Conjunction:
 * $\neg \neg \map P x$

and so by Double Negation:
 * $\map P x$

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 $\neg \exists x: \map S x$.

From the definition of logical conjunction, it follows that:
 * $\neg \paren {\exists x: \map S x \land \map P x}$

Similarly from the definition of logical conjunction, it follows that:
 * $\neg \paren {\exists x: \map S x \land \neg \map P x}$

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 $\exists x: \map S x$.

Also defined as
Some sources gloss over the possibility of $\map S x$ being vacuous and merely report that the particular affirmative and particular negative are subcontrary.