Question: For events $A_1, A_2, \ldots, A_n$ consider the $2^n$ equations $P(B_1\cap\ldots\cap B_n)=P(B_1)\ldots P(B_n)$ with $B_i=A_i$ or $B_i=A_i^c$ for each $i$. Show that $A_1,\ldots,A_n$ are independent if all these equations hold.
Note that this is different of the definition when $B_i$ is either $B_i=A_i$ or $B_i=\Omega$. I have no clue how to do this.
Thanks in advance.