As far as I know, the celebrated Elkik's result often quoted boil down as follows$\colon$ Let $A$ be a heneselian local ring with its maximal ideal ${\frak m}$ and choose an arbitrary ideal $I \subset {\frak m}$. Let us denote by $\widehat{A}$ the completion $\underset{n}{\varprojlim}\, A/I^n$. Then we have the following equality$\colon$ $$ \pi_1({\mathrm{Spec}}\,A \setminus V(I)) = \pi_1({\mathrm{Spec}}\,\widehat{A} \setminus V(I)). $$
Q. What is the rough idea of the proof?
For example, when $A = k[X]^{\mathrm{h}}_{(X)}$ is the henselization of $k[X]$ at $(X)$, we can use Kummer theory or Artin-Schreier theory according to the order of the Galois covering. For example, $A/({\frak{P}}(A) - A)$, where ${\frak{P}}$ is just $p$-th power map, is the module filtrated by $\frac{k}{X^n}$ for $n \geq 0$.