the independence property of conditional expectation

I read a proof of following property: suppose $$X$$ is a random variable on the probability space $$(\Omega,F,P)$$, $$A$$ and $$B$$ are sub $$\sigma$$-field of $$F$$, $$B$$ is independent of $$\sigma(X,A)$$, then $$E(X|\sigma(A,B))=E(X|A)$$.

Proof: denote $$E(X|A)$$ by $$Y$$, we want to show for all $$C \subset \sigma(A,B)$$,$$E(X1_C)=E(Y1_C)$$. For $$C=A' \cap B'$$ where $$A' \in A,B' \in B$$, it is trivial, we can show it also holds for the field generated by $$\{C| C=A' \cap B'\}$$. Because the monotone class generated by a field equals to the $$\sigma$$-field generated by the same field, we get the conclusion.

Question: WHY $$E(X1_C)=E(Y1_C)$$ holds on the field generated by $$\{C| C=A' \cap B'\}$$?