I don't think the corollary is true without further assumptions. 

Take $R = k[x,y,z]$ and $I = (x(y-1), y,z(y-1)$. Since $x(y-1), y,z(y-1)$ is a regular sequence, hence depth $(I,R)$ = 3. But $x(y-1),z(y-1), y$ is not a regular sequence.


I believe that the statement is that $I$ can be generated by an $M$-sequence. If $M = R$, then it is Exercise 16.9 in the same section in Matsumura's book.