I was wondering how one would negate an existential quantifier over a logical conjunction. As an example, suppose that I have the statement: "There exists a car that is white and doesn't use diesel".
I am recognizing this as saying: $\exists x \in C: \ (P(x) \land Q(x))$, where $C$ is the set of all cars, and $P$ is the statement that "a car is white" while $Q$ is the statement "doesn't use diesel".
My question is if the statement can be represented using these quantifiers, and if so, what the negation would be. Would the negation be:
$$ \neg \exists x \in C: \ (P(x) \land Q(x)) \implies \forall x \in C \ : \ \neg P(x) \lor \neg Q(x) $$
In other words, the negation is: "All cars are either not white or use diesel"? Thanks.