I am trying to understand a proof in the book "Backward Stochastic Differential Equations" by Jianfeng Zhang, however twice the following Problem is used. I have actually no idea how to prove this problem. Has anyone an idea? Thanks in advance.
Let $\mathcal{G}\subset \mathcal{F}$ be a sub $\sigma$-algebra. Let $X$ be measurable w.r.t. $\mathcal{G}$ and $\phi:\mathbb{R}^d \times \Omega \to \mathbb{R}$ be bounded and $\mathcal{B}(\mathbb{R}^d)\times\mathcal{F}$-measurable. Assume, for each $x\in \mathbb{R}^d$, that $\phi(x,{}\cdot{})$ is independent of $\mathcal{G}$. Show that $$\mathbb{E}[\phi(X,\omega)\mid\mathcal{G}]=\mathbb{E}[\phi(x,{}\cdot{})]\mid_{x=X}.$$