Is this correct: $E(Y\mid X,U)=E(Y\mid X,U=0)P(U=0)+E(Y\mid X,U=0)P(U=1)$ if $U$ is a discrete taking values $0, 1$

Question

Is this correct: $E(Y\mid X,U)=E(Y\mid X,U=0)P(U=0)+E(Y\mid X,U=0)P(U=1)$ if $U$ is a discrete taking values $0, 1$. Further more, can I do this, with the same $U$: Let $X=U+V$ $E(Y\mid X)=E(Y\mid X, U=0)P(U=0)+E(Y\mid X, U=1)P(U=1)$?