Negation of Propositional Function in Two Variables

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


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

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 $\map P {x, y}$


\(\ds \neg \forall x: \, \) \(\ds \exists y\) \(:\) \(\ds \map P {x, y}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists x: \, \) \(\ds \neg \exists y\) \(:\) \(\ds \map P {x, y}\) Denial of Universality
\(\ds \leadstoandfrom \ \ \) \(\ds \exists x: \, \) \(\ds \forall y\) \(:\) \(\ds \neg \map P {x, y}\) Denial of Existence