I'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m)$, where $m$ is the maximal ideal, and a finitely generated non-zero $R$-module $M$ such that its projective dimension $\operatorname{pd}_R M$ is finite then the following formula holds: $\operatorname{pd}_R M + \operatorname{depth} M= \operatorname{depth} R.$
I remind you that the quantity $\operatorname{depth} M$ in general is defined to be the length of a maximal regular sequence $\bar{x}=x_1,\dots,x_n$ on $M$ in $m$ (i.e. $\bar{x}$ is the maximal sequence of elements in $m$ such that (1) $M/(\bar{x})M\neq 0$ and (2) $x_j$ is a non-zero divisor in $M/(x_1,\dots,x_{j-1})M$).
In the proof of this theorem (proven by induction) it is used the fact that the projective dimension $\operatorname{pd}_R M$ is equal to $\operatorname{pd}_{R/xR}M/xM$, where $x$ is not a zerodivisor on $M$ nor on $R$, but I cannot understand why this should be true... Can you help me?