J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator):
Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with eigenvalue $E_n:$ $$H|n> = E_n|n>;$$ then $|n>$ is also a parity eigenket.
Then as a proof he uses some fact that is specific to the parity operator.
But isn't this trivially true even if we replace $\pi$ with any other operator if $H$ has a nondegenerate spectrum? is there a case where $H$ commutes with an operator and has nondegenerate eigenket, yet the eigenket is not an eigenket of the other operator?
Is this so if $H$ is not diagonalizable?
Why does this deserve the status of a "theorem"?