Satisfiable iff Negation is Falsifiable

Theorem
Let $\mathbf A$ be a WFF of propositional logic.

Then $\mathbf A$ is satisfiable its negation $\neg \mathbf A$ is falsifiable.

Necessary Condition
Let $\mathbf A$ be satisfiable.

Then there exists a boolean interpretation $v$ of $\mathbf A$ such that:


 * $\map v {\mathbf A} = \T$

Hence, by definition of the boolean interpretation of negation:


 * $\map v {\neg \mathbf A} = \F$

It follows that $\neg \mathbf A$ is falsifiable.

Sufficient Condition
Let $\neg \mathbf A$ be falsifiable.

Then there exists a boolean interpretation $v$ of $\neg \mathbf A$ such that:


 * $\map v {\neg \mathbf A} = \F$

Hence, by definition of the boolean interpretation of negation:


 * $\map v {\mathbf A} = \T$

It follows that $\mathbf A$ is satisfiable.