$$P(x) \equiv (x \neq 1 \land (\forall y, z \in \mathbb{N}): (x = yz \rightarrow (y = 1 \lor y = x))), \forall x \in \mathbb{N}.$$
Question statement: If $P(x)$ is true, what can be said about $x$?
In this case, $\mathbb{N}$ includes the number $0$, which means if $P(x)$ is true then $x$ must be prime or equal to $0$? Or is there any other significant property of $x$ that I am missing out from the statement? Thank you!
If $P(x)$ holds, then $x$ is a prime number, but $x \neq 0$ by the counterexample you just mentioned. Just a comment on the syntax of the expression you've written; I would avoid using $=$ after $P(x)$ to denote "stands for", since the equality symbol has already a well-defined meaning when interpreted as standard equality of natural numbers. Instead, you can use the symbol $\equiv$.