For any random variables x,y,z determining probability distributions $p_x, p_y, p,z$ show that
$H(\overline{p}_{x|} \overline{p}_y, \overline{p}_z) \geq H(\overline{p}_{x|} \overline{p}_y) - H(\overline{p}_{z|} \overline{p}_y) \geq H(\overline{p}_{x|} \overline{p}_y)-H(\overline{p}_z)$
I understand that $H(\overline{p}_{x|} \overline{p}_y, \overline{p}_z) = H(\overline{p}_x,\overline{p}_y,\overline{p}_z) - H(\overline{p}_y, \overline{p}_z)$ but I'm having trouble with relating $H(\overline{p}_x,\overline{p}_y,\overline{p}_z)$ term to the ones I need in the inequality.
$\begin{align*} H(\bar{p}_x | \bar{p}_y, \bar{p}_z) &= H(\bar{p}_x, \bar{p}_y, \bar{p}_z) - H(\bar{p}_y, \bar{p}_z) \\ &=H(\bar{p}_x, \bar{p}_y, \bar{p}_z) - H(\bar{p}_z | \bar{p}_y) - H(\bar{p}_y) \\ &\geq H(\bar{p}_x | \bar{p}_y) + H(\bar{p}_y) - H(\bar{p}_z | \bar{p}_y) - H(\bar{p}_y) \\ &= H(\bar{p}_x | \bar{p}_y) - H(\bar{p}_z | \bar{p}_y) \\ &\geq H(\bar{p}_x | \bar{p}_y) - H(\bar{p}_z) \end{align*}$
The first inequality follows from $H(\bar{p}_X, \bar{p}_y, \bar{p}_z) \geq H(\bar{p}_x, \bar{p}_y) = H(\bar{p}_x | \bar{p}_y) + H(\bar{p}_y)$. The second inequality is the fact that conditioning can't increase entropy.