Lovasz-Local-Lemma, question in the proof (independence of complements)

Question

In the common proof of the Lovasz-Local-Lemma (e.g. from Wikipedia), when bounding the numerator, there is always this equality $$P(A \vert \bigcap_{B \in S_{2}} \bar{B}) = P(A)$$, basically saying that $A$ and $\bigcap_{B \in S_{2}} \bar{B}$ are independent.

This is derived from the condition that $A$ is mutually independent of $S_{2}$, but how do you show this?

In other words, how can you show this conclusion:

$A$ is mutually independent of $\{B_{i} \vert 1 \le i \le n\}$ $\Rightarrow$ $A$ is mutually independent of $\{\bar{B_{i}} \vert 1 \le i \le n\}$

Thank you.

Answer 1

Ok, please don't respond - I finally came up with a solution myself.

Since I'm short on time atm, I will post my answer on Thursday here in case anyone should ever have the same problem.

Edit: And I will delete this.