Universal Affirmative and Negative are both False iff Particular Affirmative and Negative are both True

Theorem
Consider the categorical statements:

Then:
 * $\mathbf A \left({S, P}\right)$ and $\mathbf E \left({S, P}\right)$ are both false


 * $\mathbf I \left({S, P}\right)$ and $\mathbf O \left({S, P}\right)$ are both true.
 * $\mathbf I \left({S, P}\right)$ and $\mathbf O \left({S, P}\right)$ are both true.

Necessary Condition
Let $\mathbf A \left({S, P}\right)$ and $\mathbf E \left({S, P}\right)$ both be false.

Hence $\mathbf I \left({S, P}\right)$ and $\mathbf O \left({S, P}\right)$ are both true.

Sufficient Condition
Let $\mathbf I \left({S, P}\right)$ and $\mathbf O \left({S, P}\right)$ both be true.

Hence $\mathbf A \left({S, P}\right)$ and $\mathbf E \left({S, P}\right)$ are both false.