If a theorem is true for a sub/super-martingale does it also hold for a martingale?

Question

If a theorem is true for a sub/super-martingale does it also hold for a martingale? Since a sub-martingale has the characterestic property $\mathbb{E}[X_n|\mathcal{F}_m] \geq X_m$ (with $m\leq n$) which includes the case of equality and hence also the martingale case?