TITLE: Equivalent expressions for $\lnot (\forall x)\exists y A $
QUESTION [6 upvotes]: I need equivalent expressions for: $\lnot (\forall x)\exists y A$

$\exists x \lnot (\exists y A)$
$\exists x \lnot (\exists y) A$ The same as (1) I think
$\exists x \exists y \lnot A $

Which one is OK?  And what would be the result for the negation: $\lnot(\lnot (\forall x)\exists y A)$?

$\forall x \lnot (\exists y A) $

Is that OK?
In fact, what I want is $\lnot (\forall x)\exists y A$ without any negated quantifier, if this is possible.

REPLY [3 votes]: Since this appears to be homework, I will offer an approach to the solution, rather than a solution. You should consider two questions:

How can you rewrite $\lnot \forall x\,A$ in terms of $\exists$?
How can you rewrite $\lnot \exists x\,A$ in terms of $\forall$?