Negating multi-layered statement regarding a formal language

Question

I'm having trouble negating a nested statement.

Let $\Sigma$ be an alphabet, $L\subseteq\Sigma^{*}$ a language and $n\in\mathbb{N}$ a natural number.

For all words $x\in{}L$ with $|x|\geq{}n$ there exists a partition $x=abc$ with $|ab|\leq{}n$ and $|b|\geq{}1$, so that for all $i\geq{}0$, $ab^{i}c$ is in $L$.

Suggested negation:

There exists a word $x\in{}L$ with $|x|\geq{}n$ for which all partitions are $x=abc$ with $|ab|\leq{}n$ and $|b|\geq{}1$, so that for all $i\geq{}0$, $ab^{i}c$ is not in $L$.

This doesn't sound quite right and I would appreciate suggestions for a correct negation.

Answer 1

The statement has the form :

$\forall x \ \exists p \ \forall i \ P(x,p,i)$.

Thus, its negation must be :

$\exists x \ \forall p \ \exists i \ \lnot P(x,p,i)$

where $\lnot P$ is :

"$ab^ic$ is not in $L$".