Let $(\Omega,\mathcal{A},P)$ be a probability space. Given $A,B,C \in \mathcal{A}$ one can easily verify that $$ P[A\cap B \ |\ C] = P[A | B \cap C] P[B \ | \ C] . $$
Given a $\sigma$-sub algebra $\mathcal{B}$ of $\mathcal{A}$ and two $X,Y \in L^2(\Omega,\mathcal{A},P)$, can one more generally derive a formula for $$ E[XY \ | \ \mathcal{B} ] $$ involving $$ E[X,\sigma(Y,\mathcal{B})] \ \mbox{and} \ E[Y,\mathcal{B}] ? $$.