Proof regarding logic and negation of quantifiers I've read about the following task, but don't know how to prove it:
Proof that $\neg(\forall  x ( V(x)\rightarrow F(x))\iff \exists x( V(x) \land \lnot F(x)) $.
Maybe we start by proving "$\Leftrightarrow$" by proving the contraposition:
$\neg\exists x (V(x) \land \lnot F(x)) \Rightarrow \forall  x ( V(x)\rightarrow F(x))$
Now we can make a contradiction by assuming:
$\exists x (V(x)\land \lnot F(x))$  but how do I move on now?

Here is an approach to proving the biconditional using a Fitch-style proof checker. Using a proof checker makes sure that I am following the rules.

I used the following rules: change of quantifiers (CQ), universal elimination (∀E), universal introduction (∀I), De Morgan's laws (DeM), negation elimination (¬E), negation introduction (¬I), disjunctive syllogism (DS), double negative elimination (DNE), conditional introduction (→I), conditional elimination (¬E), indirect proof (IP), existential elimination (∃E) and biconditional introduction (↔I).

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
