**Proof.**
We will prove the lemma by induction on $d = \dim (\mathcal{O}_{X, x})$.

An uninteresting case is $d = 1$ since in that case the morphism $f$ is étale at $x$ by assumption. Assume $d \geq 2$.

We can base change by $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to Y$ without affecting the conclusion of the lemma, see Morphisms, Lemma 29.36.17. Thus we may assume $Y = \mathop{\mathrm{Spec}}(A)$ where $A$ is a regular local ring and $y$ corresponds to the maximal ideal $\mathfrak m$ of $A$.

Let $x' \leadsto x$ be a specialization with $x' \not= x$. Then $\mathcal{O}_{X, x'}$ is normal as a localization of $\mathcal{O}_{X, x}$. If $x'$ is not a generic point of $X$, then $1 \leq \dim (\mathcal{O}_{X, x'}) < d$ and we conclude that $f$ is étale at $x'$ by induction hypothesis. Thus we may assume that $f$ is étale at all points specializing to $x$. Since the set of points where $f$ is étale is open in $X$ (by definition) we may after replacing $X$ by an open neighbourhood of $x$ assume that $f$ is étale away from $\overline{\{ x\} }$. In particular, we see that $f$ is étale except at points lying over the closed point $y \in Y = \mathop{\mathrm{Spec}}(A)$.

Let $X' = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$. Let $x' \in X'$ be the unique point lying over $x$. By the above we see that $X'$ is étale over $\mathop{\mathrm{Spec}}(A^\wedge )$ away from the closed fibre and hence $X'$ is normal away from the closed fibre. Since $X$ is normal we conclude that $X'$ is normal by Resolution of Surfaces, Lemma 54.11.6. Then if we can show $X' \to \mathop{\mathrm{Spec}}(A^\wedge )$ is étale at $x'$, then $f$ is étale at $x$ (by the aforementioned Morphisms, Lemma 29.36.17). Thus we may and do assume $A$ is a regular complete local ring.

The case $d = 2$ now follows from Lemma 58.28.2.

Assume $d > 2$. Let $t \in \mathfrak m$, $t \not\in \mathfrak m^2$. Set $Y_0 = \mathop{\mathrm{Spec}}(A/tA)$ and $X_0 = X \times _ Y Y_0$. Then $X_0 \to Y_0$ is étale away from the fibre over the closed point. Since $d > 2$ we have $\dim (\mathcal{O}_{X_0, x}) = d - 1$ is $\geq 2$. The normalization $X_0' \to X_0$ is surjective and finite (as we're working over a complete local ring and such rings are Nagata). Let $x' \in X_0'$ be a point mapping to $x$. By induction hypothesis the morphism $X'_0 \to Y$ is étale at $x'$. From the inclusions $\kappa (y) \subset \kappa (x) \subset \kappa (x')$ we conclude that $\kappa (x)$ is finite over $\kappa (y)$. Hence $x$ is a closed point of the fibre of $X \to Y$ over $y$. But since $x$ is also a generic point of this fibre, we conclude that $f$ is quasi-finite at $x$ and we reduce to the case of purity of branch locus, see Lemma 58.21.4.
$\square$

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