Equivalent definition of maximal Cohen-Macaulay modules over a Gorenstein local ring

Question

$ \newcommand{\Ext}{\mathop{\rm Ext}\nolimits} \newcommand{\depth}{\mathop{\rm depth}\nolimits} \newcommand{\dim}{\mathop{\rm dim}\nolimits} $

A module $M$ is a maximal Cohen-Macaulay module over the ring $R$ if $\depth M = \dim R$, where $\dim R$ denotes the Krull dimension of $R$. I've been told that if $R$ is a Gorenstein local ring, then $M$ is a maximal Cohen-Macaulay module if and only if $\Ext^i(M,R)=0$ for all $i \geq 1$. (This is taken as the definition here.)

How can this be proved? And does anyone have a source where this has been proved? Is it true for Gorenstein local rings only, or also for Gorenstein (not necessarily local) rings too?

I primarily work with commutative rings and finitely generated modules.