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

Theorem
Consider the categorical statements:

Then:
 * $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ are both false


 * $\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ are both true.
 * $\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ are both true.

Necessary Condition
Let $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ both be false.

Hence $\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ are both true.

Sufficient Condition
Let $\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ both be true.

Hence $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ are both false.