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We may assume that $\phi'$ is standard. Let $a\in A$ be a nonzero nonunit. Since $\mathfrak{m}$ is a proper ideal, the leading coefficient of $\phi'_a$ cannot vanish mod $\mathfrak{m}$. Thus $\phi_a'$ and $\phi_a$ have the same degree, so $\phi'$ and $\phi$ have the same rank.