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Suppose that $X$ is an integrable r.v. on probability measure space $(\Omega,F,P)$. Show that $E[X|G_1,G_2]=E[X|G_1]$, where $G_1,G_2$ are sub $\sigma$-algebras and $\sigma(X,G_1)$ and $G_2$ are independent

Related: Conditional expectation on more than one sigma-algebra

I've got a lot of help from the above question posted on MSE, but even with those I'm not sure how I'd go about proving

$$ \int_{A\cap B} E[X|G_1]=\int_{A\cap B} X $$

for every $A\in G_1$, $B\in G_2$. In particular, I cannot see how relation $P(A\cap B)=P(A)P(B)$ translates to the integrals. What'd be the best way to go about this one?

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user160738
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    Hint: Note that $$ \int_{A\cap B} E(X\mid G_1)dP=E(\mathbf 1_A\mathbf 1_BE(X\mid G_1))\qquad \int_{A\cap B} XdP=E(\mathbf 1_A\mathbf 1_BX) $$ and look for independences. – Did Nov 16 '15 at 06:46

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