By definition I have learned that:
$ (\exists x)B \mbox{ stands for }(\lnot((\forall x)(\lnot B))) $
But in a logical proof I need the identity:
$(\forall x) B \mbox{ stands for } (\lnot((\exists x)\lnot B)) $ Can I just derive this identity or need I prove it (with the axioms, modus ponens, generalization or the deduction theorem)?
In the last case, it is hard. Since it seems so trivial, I can't think of a clever way to prove it, other than claiming it is trivial, but in logic everything needs explanation..
$$(\forall x) B = \neg(\neg((\forall x) \neg(\neg(B)))) = \neg((\exists x) \neg B))$$