I am interested in how one would formally prove:
$\lnot \forall x, P(x) \iff \exists x, \lnot P(x)$
I realize that it's basically saying that:
$\lnot(P(x_0) \land P(x_1) \land ... \land P(x_n)) \iff \lnot P(x_0) \lor \lnot P(x_1) \lor ... \lor \lnot P(x_n)$
Which is an "intuitive" proof assuming we already accept De Morgan's, but I am curious if there is a formal way to prove it (e.g. Fitch-style).