Let $R$ be a regular Noetherian local ring of dimension $d$ with maximal ideal $m\subset R$. For elements $a_1,\dots,a_r\in m$ show that the quotient $R/(a_1,\dots,a_r)$ is regular of dimension $d-r$ if and only if the residue classes $\bar a_1,\dots,\bar a_r\in m/m^2$ are linearly independent over $k:=R/m$.
($\Rightarrow$) $R$ is regular implies $\operatorname {dim}_k m/m^2=d$. Assume also that $R/(a_1,\dots,a_r)$ is regular of dimension $d-r$: then $\frac{m}{m^2+( a_1,\dots, a_r)}\cong \frac{m/m^2}{(\bar a_1,\dots,\bar a_r)}$ is a $k$-vector space of dimension $d-r$, so the elements $\bar a_1,\dots,\bar a_r$ span a subspace in $m/m^2$ of dimension $r$.
($\Leftarrow$) If conversely $\bar a_1,\dots,\bar a_r$ span a subspace of dimension $r$, then $\operatorname {dim}_k \frac{m/m^2}{(\bar a_1,\dots,\bar a_r)}=d-r$. It is known that if exist elements $a_{r+1},\dots, a_d\in m$ such that $a_1,\dots,a_d$ generate $m$, then $\operatorname{dim}R/(a_1,\dots ,a_r)=d-r$. So take a basis of the $k$-vector space $m/m^2$, that can be assumed of the form $\bar a_1,\dots,\bar a_r,\bar c_{r+1},\dots \bar c_d$, where $\bar c_{r+1},\dots \bar c_d$ are the residue classes of some $c_{r+1},\dots c_d\in m$ (as every linearly independent subset of a vector space can be extended to a basis). By Nakayama's lemma $a_1,\dots,a_r,c_{r+1},\dots c_d$ generate $m$.
Do you think is correct? It looks quite simple so I suspect that I assumed something that isn't generally true without seeing it, but I checked as well as I could every passage.