Is a Gröbner basis after substitution still a Gröbner basis?

Question

Suppose we have an ideal $I = (g_1,\dots,g_k)\subseteq k[x_1,\dots,x_n]$ where $g_1,\dots,g_k$ is a Gröbner basis. Let $(c_1,\dots,c_n)\in V(I)$ be a point and consider the ideal $I_n = (\overline{g_1},\dots,\overline{g_k})\subseteq k[x_1,\dots,x_{n-1}]$ where $\overline{g_i}= g_i(x_n=c_n)$. Is it true that $\overline{g_1},\dots,\overline{g_k}$ is a Gröbner basis of $I_n$? I tried proving it using the Buchberger criterion and also just from the definition but I failed. I feel like I need another condition on the monomial ordering, for example lexi order $x_1>\dots>x_n$ should be enough for the statement to be true but I am not sure.