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 $\map P {x, y}$

Proof

 $\, \displaystyle \neg \forall x: \,$ $\displaystyle \exists y$ $:$ $\displaystyle \map P {x, y}$ $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists x: \,$ $\displaystyle \neg \exists y$ $:$ $\displaystyle \map P {x, y}$ Denial of Universality $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists x: \,$ $\displaystyle \forall y$ $:$ $\displaystyle \neg \map P {x, y}$ Denial of Existence

$\blacksquare$