Depth of ideals in a commutative ring

Question

Let $I,J\subset S=k[x_1,\dots,x_n]$ be two monomial ideals and $k$ a field. If every regular element on $S/J$ is also a regular element on $S/I$ is it true that $\operatorname{depth}_S S/I\ge \operatorname{depth}_S S/J$ ?