Discovering the Useful Denial for Predicates P(x) and Q(x) in Universe U

[ver_mathstats]

Let U be universe under consideration, and let P(x) and Q(x) be predicate with free variable x. Find a useful denial for (∀x∈U)(P(x)⇒Q(x))

Then my answer is:

¬(∀x∈U)(P(x)⇒Q(x))

(∃x∈U)¬(P(x)⇒Q(x))

¬(P(x)⇒Q(x))

¬P∨Q

I'm unsure if I am on the write path when it comes to finding the useful denial, is this how to do it?

Thank you.

 

[fresh_42]

Your last step is wrong.
$$
\lnot \,(P(x) \Longrightarrow Q(x)) \Longleftrightarrow \exists \,x_0 \in U \, : \,P(x_0) \wedge \lnot \,Q(x_0)
$$