Does “double” conditional zero expectation follow from two conditional zero expectations?

If we have $E(x|y)=E(x|z)=0$, does it follow that $E(x|y,z)=0$?

• Your second conditional expectation specified no value. Please check my edit. – Christoph Hanck Jan 18 '18 at 9:49

No. Let $E[z|y] = -y$, $E[y|z]= -z$ and $x = y +z$. Then $E[x|y] = y + E[z|y] = y -y = 0$, $E[x|z] = E[y|z] + z = -z + z = 0$ but $E[x|y,z] = y + z$.