I am trying to understand how to rewrite the variance of regression coefficients in terms of the variance inflation factor. To be specific If we have a regression with design matrix X, I know that we have:

$$
\widehat{\operatorname{var}}(\hat{\beta}_j) = s^2 (X^T X)^{-1}
$$

and my goal is to show that:

$$
\widehat{\operatorname{var}}(\hat{\beta}_j) = \frac{s^2}{(n-1)\widehat{\operatorname{var}}(X_j)}\cdot \frac{1}{1-R_j^2}
$$

where $R_j$ is the multiple r-squared for the regression between $X_j$ and the other predictors and $s^2$ is the residual standard error. 

I have seen a proof on Wikipedia (https://en.wikipedia.org/wiki/Variance_inflation_factor#Definition) but would appreciate some help, either in understanding this proof or in providing a different proof or link to a reference to read.