Converting ∃ to ∀ and vice versa I'm having some trouble getting my head around the conversion of quantifiers.
For instance, I know that $\forall x  \,F \,\equiv\, \neg\exists x \, \neg F$ and conversely. $\exists x  \,F \,\equiv\, \neg\forall x \, \neg F$ 
However, what if I have $\neg\forall x\forall y \,F $? What's the logical equivalence for this? 
Furthermore, is $\neg\forall x\forall y \,F \equiv \neg\forall x\neg\forall y \,F$, or are they 2 difference things? 
If they are difference, then I believe $\neg\forall x\forall y \,F \equiv \exists x\neg\exists y \,\neg F$, but I am not sure about how to generally convert.
Any insight is much appreciated. I've tried googling but most results only come up with the simple case discussed above, that I already know.
 A: No, $\neg\forall x\forall y\, F$ is not the same as $\neg\forall x\neg \forall y\, F$. For example if we take $F$ to mean that $x$ and $y$ are friends, then $\neg\forall x\forall y\, F$ means "it is not the case that everyone is friends with everyone", whereas $\neg \forall x\neg \forall y\, F$ means "there is nobody who does not have any friends", and it should be easy to imagine situations where one of these is true and the other is not.
On the other hand $\neg \forall x\forall y\, F$ is equivalent to $\exists x\exists y \,\neg F$.
It's usually more intuitive to remember these rules in the form
$$ \neg \forall x\, F \equiv \exists x\,\neg F \qquad\qquad
\neg \exists x\, F \equiv \forall x\,\neg F $$
or in words:

You can move a negation through a quantifier if you flip the quantifier to the other type as you do it.

This way, when you have a whole string of quantifiers with a negation at one end, you can shift the negation to the other when while flipping each of the quantifier. So, for example,
$$ \neg \forall x \exists y \exists z \forall w \, F \equiv
\exists x\forall y\forall z\exists w\,\neg F $$
