How to prove that the Myhill-Nerode equivalence classes for L are the same as for its complement? Given language $L$, I want to show that its Myhill-Nerode equivalence classes are the same as for its complement $\overline{L}$.

I am thinking of constructing a DFA $M$ for the Language $L$ so the DFA for the language of its complement will be just changing the final states to non final states but the states will be same. So the classes do remain same? How do I put this in a more formal way?
Note that the Myhill-Nerode equivalences are also defined for non-regular languages, so constructing a finite state automaton will not help.

What is the definition of such a class? $x \sim_L y$ iff $xz\in L$ whenever $yz\in L$ (for any string $z$). How does that change when we consider the complement of $L$? We use basic logic: $P\Leftrightarrow Q $ iff $\lnot P\Leftrightarrow \lnot Q $.

$x \sim_L y$
iff  $xz\in L$ whenever $yz\in L$ (for any string $z$)
iff  (not $xz\in L$) whenever (not $yz\in L$) (for any string $z$)
iff  $xz\in \bar L$ whenever $yz\in \bar L$ (for any string $z$)
iff  $x \sim_{\bar L} y$.