Conditional probability over multiple conditions, A is independent of D given C. Does independence hold if I add one more event in the condition?

Question

If $A$ is independent of $D$ given $C$, does that mean that $A$ will also be independent of $D$ given $B$ and $C$? Basically, does

$P(A|C, D) = P(A|C)$

imply

$P(A|B,C,D) = P(A|B,C)$

To me, it seems to be obvious that adding an extra condition should not change the independence. But is there any way to prove it? If it's not true, can someone give an example where it doesn't hold?

Answer 1

Toss a coin twice, and let $A$ be the event that the first coin lands heads, let $D$ be the event that the second coin lands heads, let $C$ be the entire sample space, and let $B$ be the event that the two coin tosses land differently. Clearly $A$ and $D$ are independent given $C$. However, it's easy to check that $P(A\mid B,C)=\frac12 $ while $P(A\mid B, C, D)=0$.

Answer 2

To me, it seems to be obvious that adding an extra condition should not change the independence.

To me it seems rather obvious that adding an extra condition may very plausibly break independence.