Let $X$ be a variety over $\mathbb{C}$, and $X^{an}$ be the analytic space associated to $X$. $X$ has Gorenstein singularity at $x \in X$ iff the local ring $\mathcal{O}_{X,x}$ is a Gorenstein ring. Is this result true:

$\mathcal{O}_{X,x}$ is a Gorenstein $\iff$ $\mathcal{O}_{X^{an},x}$ is a Gorenstein.

I want to show the following thing: Suppose $X,Y$ are varieties over $\mathbb{C}$, and $x \in X^{an}, y\in Y^{an}$. Suppose $U \subset X^{an}, V\subset Y^{an}$ are open sets in the Eucliden topology, and there exists an analytic isomorphism $f : U \to V$ sending $x$ to $y$. Then, if $X$ has Gorenstein singularity at $x$, it also has Gorenstein singularity at $y$. Certainly, if the aforementioned result holds, this result follows from it.