predicate logic negation laws, two quantifiers

Question

Is $\lnot\, \forall\ x\exists\, y \lnot\ P(x,y)$ equivalent to $\exists\, x \exists yP(x,y)$ ?

I understand the negation rule, I just need to make sure i got it right. help please

Answer 1

We have to proceed by step :

$¬∀x \ ∃y \ ¬P(x,y)$

is equivalent to :

$∃x \ ¬ ∃y \ ¬P(x,y)$

which in turn is equivalent to :

$∃x \ ∀y \ P(x,y)$.