Dirichlet Problem: symmetry of the solution with respect to the $x_1$ variable

Question

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ such that for every $(x_1,\ldots,x_n)\in\Omega$, the point $(-x_1,\ldots,x_n)$ is also in $\Omega$. Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$ be a solution of the Dirichlet problem $$ \begin{cases} \Delta u = 0\hspace{0.3cm}\text{in}\hspace{0.3cm}\Omega\\ u = \phi\hspace{0.3cm}\text{on}\hspace{0.3cm}\partial\Omega \end{cases} $$ where $\phi\in C^0(\partial\Omega)$ verifies $\phi(-x_1,\ldots,x_n)=\phi(x_1,\ldots,x_n)$. Prove that $$ u(-x_1,\ldots,x_n)=u(x_1,\ldots,x_n). $$