How to convert statement with the existential quantifier to statement with universal quantifier?
Let's give names for the atom formulas:

$n&gt;1$ will be $\phi$ and $x(n) \neq 1$ will be $\psi$
Now what we have is:
$$\exists{n} (\phi\rightarrow\psi)$$
$$\exists{n} (\neg\phi \lor\psi)$$
$$\neg\neg \exists{n} (\neg\phi \lor\psi)$$
$$\neg\forall{n} \neg(\neg\phi \lor\psi)$$
$$\neg\forall{n} (\phi \land \neg\psi)$$
$$\neg\forall{n}((n&gt;1) \land \neg(x(n)\neq 1))$$
$$\neg\forall{n}((n&gt;1) \land (x(n) = 1))$$