Suppose that $A,B,C$ are random variables such that $A$ is independent from $C$.
Does it hold that $$ \mathbb{E}[A \mid \sigma(B)] = \mathbb{E}[A \mid \sigma(B,C)]? $$
I tried to go to the conditional expectation wikipedia page https://en.wikipedia.org/wiki/Conditional_expectation and use the basic properties, but I was not succesful.
It's not true in general.
You can consider an example where $A$ is independent of $B$, and also $A$ is independent of $C$, but where $A$ is determined by $(B,C)$.
For example, let $B$ and $C$ be i.i.d. each taking values $0,1$ with probability $1/2$ each, and let $A=(B+C) \bmod 2$. Then $A$ also takes values $0,1$ w.p. $1/2$ each, and you get the pairwise independence properties mentioned above.
So $\mathbb{E}[A|\sigma(B)]=\mathbb{E}[A]=1/2$ with probability $1$, but $\mathbb{E}[A|\sigma(B,C)]=A\in\{0,1\}$ with probability $1$.