Suppose we have this Poisson equation $$ \nabla^{2}\phi(x,y,z)=\rho(x,y). $$ A solution would be of the form $$ \phi=\phi_0(x,y,z)+\phi_1(x,y) $$ where $\phi_0(x,y,z)$ is solution of the Laplace equation $\nabla^{2}\phi(x,y,z)=0$.
How can we prove that $\phi_1(x,y)$ is independent of the coordinate $z$ ?