Let $\Omega$ be a domain in $\mathbb{C}^{m},m\geq 2$ and $\mathcal{Z}$ be an analytic submanifold of codimension $2$. Also, suppose $\mathcal{Z}=\mathcal{Z}_{1}\cap\mathcal{Z}_{2}$, where $\mathcal{Z}_{i}$ is an analytic hypersurface in $\Omega$ for $i=1,2$. Now, fix $p\in\mathcal{Z}$.
Is it possible to choose local defining functions $\phi_{i}$ for $\mathcal{Z}_{i},~i=1,2$ at $p$ such that $p$ is also a regular point for the $\mathbb{C}^{2}$-valued function $(\phi_{1},\phi_{2})?$