Negation of Propositional Function in Two Variables

Theorem
Let $\map P {x, y}$ be a propositional function of two Variables.

Then:
 * $\neg \forall x: \exists y: \map P {x, y} \iff \exists x: \forall y: \neg \map P {x, y}$

That is:
 * It is not the case that for all $x$ a value of $y$ can be found to satisfy $\map P {x, y}$

means the same thing as:
 * There exists at least one value of $x$ such that for all $y$ it is not possible to satisfy $\map P {x, y}$