Given $f$ a convex function, $x_0 \in int(dom(f))$, then the subdifferential $\partial f(x_0)$ has only one point iff $f$ is differentiable in $x_0$.

Question

I am currently working on convex optimization and I have approached the mentioned statement. The right-to-left implication is straightforward but I am finding myself in trouble trying to prove the other way around. When I say differentiable, I mean in a general sense not is Gâteaux differentiable.Can anyone help with it?

pstd: I have seen the post How to show $\partial f(x) =\{\nabla f(x) \}$ for a convex differentiable function? that works on this topic. I am not so sure that the left-to-right implication is right, because to take $\phi(x)'$ is assuming that $f$ is differentiable (and so has gradient), but it is pretty much what we are trying to proof.

Any suggestion will be appreciated! (edited)