Proof regarding logic and negation of quantifiers

Question

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?

Answer 1

I don't know how you are supposed to prove this (Formal proof? Using equivalences? Some other method?), but let me do one that uses the following basic equivalences:

$\neg \forall x \varphi(x) \Leftrightarrow \exists x \neg \varphi(x)$ (Quantifier Negation)

$\varphi \rightarrow \psi \Leftrightarrow \neg \varphi \lor \psi$ (Implication)

$\neg(\varphi \lor \psi) \Leftrightarrow \neg \varphi \land \neg \psi$ (DeMorgan)

$\neg \neg \varphi \Leftrightarrow \varphi $ (Double Negation)

OK, using these, we get:

$\neg \forall x (V(x) \rightarrow F(x)) \Leftrightarrow $ (Quantifier Negation)

$\exists x \neg (V(x) \rightarrow F(x)) \Leftrightarrow $ (Implication)

$\exists x \neg (\neg V(x) \lor F(x)) \Leftrightarrow $ (DeMorgan)

$\exists x (\neg \neg V(x) \land \neg F(x)) \Leftrightarrow $ (Double Negation)

$\exists x (V(x) \land \neg F(x)) $