Let $S_{XY}$ a statistic that depends on two real-valued random variables $X$ and $Y$. We know that
$$V(S_{XY}) = V\Big(E(S_{XY}|X)\Big) + E\Big(V(S_{XY}|X)\Big) = V\Big(E(S_{XY}|Y)\Big) + E\Big(V(S_{XY}|Y)\Big)$$
Assume $E(S_{XY}|X)=0$. This implies that $V\Big(E(S_{XY}|X)\Big)=0$. One then obtains
$$V\Big(E(S_{XY}|Y)\Big) = E\Big(V(S_{XY}|X)\Big) - E\Big(V(S_{XY}|Y)\Big)$$
Thus, is it sure that $E\Big(V(S_{XY}|X)\Big) \geq E\Big(V(S_{XY}|Y)\Big)$ , so that $V\Big(E(S_{XY}|Y)\Big)\geq0$? Or my development is wrong?