Suppose a formula is looks like the following:
$\forall x_1 ... \forall x_n \alpha$
Where $\alpha$ is a formula free of quantifiers.
And if $P$ is a 1-ary relation letter, then the formula $\exists x P(x)$ is not equivalent to a formula like above.
It's clear to me that this is true intuitively by observing that a formula being true for one element in a model doesn't imply it's true for all. So how could one formalize this argument?