complex analysis – Proof of result of Schwarz reflection principle

If f is entire, and takes real values, for real z, then prove that the conjugate of f is equal to the function of the conjugate of z for all z in the complex plane.

I understand this is a result of the Schwarz reflection principle, but our complex analysis topic hasn’t once touched on reflection. Is there a way of proving this that involves other theorems? I’ve tried extrapolating from f=u+iv, but no where seems to have any clear working.
Is there a way of using the fact that f has real valued derivatives along the real axis?