Let $I, J \subset S = k[x_1,...,x_n]$ be two monomial ideals and $k$ a comm field. If for every regular element $\alpha$ from $S/J$, $\alpha$ is also a regular element for $S/I$ is it true that depth$_S S/I \geq$ depth$_S S/J$ ?

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