Universal Negative implies Particular Negative iff First Predicate is not Vacuous

Theorem
Consider the categorical statements:
 * $\map {\mathbf E} {S, P}: \quad$ The universal negative: $\forall x: \map S x \implies \neg \map P x$
 * $\map {\mathbf O} {S, P}: \quad$ The particular negative: $\exists x: \map S x \land \neg \map P x$

Then:
 * $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$


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

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

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

Let $\map {\mathbf E} {S, P}$ be true.

As $\map {\mathbf E} {S, P}$ is true, then by Modus Ponendo Ponens:
 * $\neg \map P x$

From the Rule of Conjunction:
 * $\map S x \land \neg \map P x$

Thus $\map {\mathbf O} {S, P}$ holds.

So by the Rule of Implication:
 * $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$

Necessary Condition
Let $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$.


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

that is, $\map S x$ is vacuous.

From De Morgan's Laws: Denial of Existence:
 * $\forall x: \neg \map S x \dashv \vdash \neg \exists x: \map S x$

it follows that $\forall x: \map S x$ is false.

From False Statement implies Every Statement:
 * $\forall x: \map S x \implies \neg \map P x$

is true.

So $\map {\mathbf E} {S, P}$ holds.

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

Then by the Rule of Conjunction:


 * $\neg \paren {\exists x: \map S x \land \neg \map P x}$

That is, $\map {\mathbf O} {S, P}$ does not hold.

So $\map {\mathbf E} {S, P}$ is true and $\map {\mathbf O} {S, P}$ is false.

This contradicts $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$ by definition of implication.

Thus $\exists x: \map S x$ must hold.

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