Let $R$ be a $d$-dimensional Noetherian regular local $k$-algbera ($k$ any field of char($k$) = 0, $d \geq$ 2). Let $x, y$ be a part of regular system of parameter for $R$. Let $I = (x, y)$ be a ideal generated by $x$ and $y$ and $\hat{R}$ denote the completion of $R$ with respect to $I$.

Is it true that $\hat{R} = \frac{R}{I}[[x, y]]$?

I know the very special case of it is true when $d = 2$ and $I$ is the maximal ideal. The above statement seems correct but I am not entirely sure. Any help would be great.