Show that a reparametrization of a parametrized curve is regular iff that curve is regular

Show that a reparametrization of $\widetilde{c}$ of a parametrized curve $c$ is regular if and only if $c$ is regular.

How do I go about answering this question? I think it's essentially asking me to show that a reparametrization of a regular curve is regular. I believe $c$ is regular if $c^{'}$ ≠ 0.

(Sorry if my formatting is bad. I'm new and I'm trying to follow a MathJax tutorial)

We are given a curve $c:I\rightarrow \mathbb{R}^{n}$. Since $\overline{c}:I'\rightarrow \mathbb{R}^{n}$ is a reparametrization of $c$, by definition there is a smooth bijection $p:I'\rightarrow I$ with nowhere vanishing derivative, such that $\overline{c}=c\circ p:I'\rightarrow \mathbb{R}^{n}$.
By the chain rule, we have $\overline{c}'(t)=c'(p(t))p'(t)$, where $p'(t)$ is nonzero. So this equation shows that $\overline{c}'(t)$ is nonzero if and only if $c'(p(t))$ is nonzero. Hence $\overline{c}$ is regular if and only if $c$ is regular.