is the $\forall x\neg P(x) = \neg \forall xP(x)$ true?

Question

I figured out this statement,

$\forall x\neg P(x) = \neg \forall xP(x)$

Can we say that these two are equal? According to me they are same but if not what would be the correct statement and if possible can you also mention an example.

Answer 1

Moving a $\lnot$ past a quantifier switches the quantifier:

$\forall x\lnot P(x) = \lnot\exists x P(x)$ -- "all apples are non-helicopters" is the same as "helicopter apples don't exist"

$\lnot\forall x P(x) = \exists x \lnot P(x)$ -- "not all people are mathematicians" is the same as "non-mathematicians exist"