Show $B= \{z \mid (\exists x)\; P(x,z)\}$ is a recursive enumerable set

Let $B = \{z \mid (\exists x)\; P(x,z)\}$ and $P$ be a computable predicate. Show $B$ is a recursive enumerable set.

My attempt

As $P$ is a computable predicate then there is a program that computes it, therefore $B= \{z \mid (\exists x)(\exists t)\;\text{STP}^{(1)}(x,z,t)\} \Rightarrow B= \{z \mid \Phi(x)\downarrow\} = W_z$ and so $B$ is a recursive enumerable set.

Further info

$\text{STP}^{(n)} (x_1, \ldots, x_n, y,t)$ is a predicate that is true if the program number $y$ halts after $t$ or fewer steps on inputs $x_1, \ldots, x_n$.

Note: please note this is the first time I ever try to solve this kind of exercises, so even if I got everything wrong and nothing makes sense, every nudge in the right direction is really welcome.

Hint, do you know the argument for the countability of the rational numbers?

Just enumerate every pair $a,b$ of natural numbers and output the $b$ if $P(a,b)$ is true.

Other way, can you construct a machine that given a $b$, answers YES if there is an $a$ such that $P(a,b)$ is true?

StackExchange Computer Science Q#14129, answer score: 2