Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.

I need help to write a rigorous proof. Thank you

We have $f_{xx}=\partial_{xx} u(x,y)+i\partial_{xx}v(x,y)$ and the same for the derivative in $y$, hence $\Delta f(x,y)=\Delta u(x,y)+i\Delta v(x,y)$. As $u$ and $v$ admit *real* values, so does $\Delta u$ and $\Delta v$, hence $\Delta u=0=\Delta v$.

As $\Delta\bar f(x,y)=\Delta u(x,y)-i\Delta v(x,y)$, we get that $\bar f$ is harmonic if (and only if) so is $f$.