Notation for there exists and let I sometimes encounter proofs as the following:

$\exists n\in\mathbb N$ such that …. Then $n$…

But the fact in that $n$ is not defined in the second sentence. A correct proof would contain instead:

$\exists n\in\mathbb N$ such that …. Let such a $n$. Then $n$…

Is there a notation to mean "there exists and let"?
 A: You are asking about notation for "there exists and let". Note that the first part of this, "there exists," is a statement telling how something is, while "let" is an imperative telling the reader what to do. They have different roles in the proof and they have somewhat different formal languages.
The formal notation of statements contains things like quantifiers ($\forall,\ \exists$) and implications ($\implies$), while the formal notation for a proof rather is a deduction tree containing statements:
$$
\dfrac
{
  \begin{matrix}\\A \lor B\end{matrix}
  \quad
  \begin{matrix}\\ [A]\end{matrix}
  \quad
  \dfrac{[B] \quad B \rightarrow A}{A}
}
{
  A
}
$$
(The example proves $A$ from $A \lor B$ and $B \rightarrow A.$)
In a proof, "there exists" will be true either by some definition or by some other theorem or lemma, and the text would read "By definition/theorem $X,$ there exists $x$ such that $P(x).$ Take such an $x.$ Then ..." Here's a deduction tree for that:
$$
\dfrac
{
  \begin{matrix}\\ \triangledown \\ \hline \exists x\ P(x)\end{matrix}
  \quad
  \begin{matrix} [P(x)] \\ \vdots \\ Q \end{matrix}
}
{
  Q
}
$$
where $\triangledown$ denotes "definition/theorem $X$" and $\vdots$ that there is some deduction tree deriving $Q$ from $P(x).$ The left part above the horizontal line corresponds to "By definition/theorem $X,$ there exists $x$ such that $P(x),$ and the right part corresponds to "Take such an $x.$ Then $P(x)$ so ... Thus $Q(x).$"
A: In $(\exists x \in X)R(x)$ in $R(x)$ $x$ is actually defined, because it is same as $(\exists x) (x \in X \land R(x))$.
Addition.
If we consider $(\exists x \in X)R(x) \land S(x)$, then it same as $(\exists y \in X)R(y) \land S(x)$.
A: I agree with your concern about $n$ not being defined in the second sentence of the first quote. Because the scope of $\exists$ would be to the end of the logical statement/sentence, and not extend/bind $n$ beyond that.
And while you'd be understood if you wrote "There exists" in words as suggested by TheSilverDoe in a comment, doing that produces a rare distinction between symbols and their usual readings, which I would prefer to avoid.
I prefer using the word "choose" (and then reserving "let" for universal quantification), as in "Choose $n\in \mathbb N$ such that $n>N+3$. Then note that $n\ge N+2$, so that...". "Choose a natural $n$ such that..." is fine as well. Or after it has been justified that something with a property exists, "So we can/may choose $n$ so that..."
You can see usage of "choose" recommended/exemplified in Doug West's mathematical writing guide The Grammar According to West in his points on expressions as units and "let x,y be".
