I am trying to better understand the situation of smooth schemes. Suppose we have a smooth affine integral $k$-scheme of finite type $X$, and let $k$ be algebraically closed (hence 'smooth' is equivalent to 'regular'). Now let $f_1, \dots, f_n \in \mathcal O_X(X)$ be a regular sequence and we also suppose that the closed subscheme $V(f_1, \dots, f_n)$ is smooth. Does it hold that for any $1 \leq i \leq n$ the closed subscheme $V(f_1, \dots, f_i)$ is smooth as well?
Thank you very much for any hints!