Any counter example for $P(A|C)=P(A),P(B|C)=P(B)$ but $P(A\cap B|C)\neq P(A\cap B)$? Let $A,B,C$ be three events, what would be an example that $P(A|C)=P(A)$ and $P(B|C)=P(B)$ do not imply $P(A\cap B|C)= P(A\cap B)$?

Consider a dice with 8 faces with numbers from $1$ to $8$ on them respectively. We roll that dice. Denote $X$ as the resulting face.
Let $A=[X=1,X=2]$, $B = [X=2,X = 3]$, $C=[X~\text{is even}]$.  Then we definitely have $$P(A|C)=P(X=2\vert C)=\frac14=P(A)$$ and $$P(B|C)=P(X=2\vert C)=\frac14=P(B).$$
But $$P(A\cap B|C) = P(X = 2\vert C) = \frac14\ne \frac18 = P(X = 2) = P(A\cap B) $$
