Laws of equivalence to show $ \forall x \in U. (P(x) \rightarrow \neg Q(x)) \equiv \neg \exists x \in U. (P(x) \land Q(x)) $

Question

Using laws of equivalence

$$ \forall x \in U. (P(x) \rightarrow \neg Q(x)) \equiv \neg \exists x \in U. (P(x) \land Q(x)) $$

show that these are logically equivalent.

Answer 1

You want:

$\forall x \in U. (P(x) \rightarrow \neg Q(x)) \equiv \neg \exists x \in U. (P(x) \land Q(x))$

So (using your notations),

$$\color{blue}\forall x \in U. (P(x) \rightarrow \neg Q(x))$$ $\equiv$ $$\color{blue}{\neg \exists} x \in U.\lnot (P(x)\color{red} \to\lnot Q(x))$$ $\equiv$ $$\neg \exists x \in U. \lnot(\color{red}\lnot P(x)\color{red} \lor \lnot Q(x))$$ $\equiv$ $$\neg \exists x \in U. (P(x) \land Q(x))$$