If two events $A$ and $B$ are conditionally independent given a third event $C$. Are they conditionally independent on $C^\complement$ as well?
i.e Does this hold? $$ P(A \cap B|C) = P(A|C) P(B|C) \implies P(A\cap B|C^\complement) = P(A|C^\complement).P(B|C^\complement) $$
I think that this is not the case. We know that for independence, the two events MUST intersect. Now I can draw a venn diagram in which $A$ and $B$ intersect only when $C$ occurs and outside $C$ they don't. Hence they won't be independent in the universe $C^\complement$. Is this a valid argument? If my argument is incorrect and the above actually holds, provide can you provide a hint for the proof? Also can you explain it intuitively(why it holds or why it doesn't hold in general)?