Excellent Local Ring: Properties

Question

Let $(R,m)$ be an excellent local ring. Denote by $\phi: R \to \hat{R}$ the canonical completion map and $f: \operatorname{Spec}(\hat{R}) \to \operatorname{Spec}(R)$ the induced map of Specs.

Let $Q \in \operatorname{Spec}(\hat{R})$ and $\phi^{-1}(Q)= f(Q)=:P \in \operatorname{Spec}(R) $.

I'm looking for a rigorous proof for following statement:

Let $S$ be the property regular, reduced, Cohen-Macaulay or Gorenstein.

Then $R_P$ has property $S$ if and only if $\hat{R}_Q$ has S.