Mutual independence is inherited by subfamilies and by replacing events with complements

With no event complemented, every intersection identity required for a subfamily is already one of the identities required for the original family.

Assume that after complementing any chosen $r$ events, every resulting subfamily is mutually independent.

Complement one further event $A_i$. For any intersection $D$ of selected events other than $A_i$, the induction hypothesis gives $\mathbb{P}(D\cap A_i)=\mathbb{P}(D)\mathbb{P}(A_i)$. Since $D\cap A_i^c=D\setminus (D\cap A_i)$, [L1] gives $\mathbb{P}(D\cap A_i^c)=\mathbb{P}(D)(1-\mathbb{P}(A_i))=\mathbb{P}(D)\mathbb{P}(A_i^c)$. Thus every subfamily remains mutually independent after $r+1$ replacements.

Induction on the number of complemented events proves the assertion for every selection. The empty intersection has probability $1$ and the empty product is $1$.