I got a little stuck on a simple proof of the following probability identity.
Given
$\mathbb{P}(A^c \cap B^c)=1-\mathbb{P}(A)-\mathbb{P}(B)+\mathbb{P}(A\cap B)$
how to prove for any set $X$,
$\mathbb{P}(X \cap A^c \cap B^c)=\mathbb{P}(X)-\mathbb{P}(X\cap A)-\mathbb{P}(X\cap B)+\mathbb{P}(X\cap A\cap B)$
Looks very intuitive; just replace the whole space by $X$. But how to prove it simply and rigorously? Thanks.