If we know A is independent of B, why isn't P(A|B,C) = P(A|C) necessarily true? Let's say we know that A is independent of B, or mathematically:
$$P(A|B) = P(A)$$
Then how come we can't say the following is necessarily true:
$$P(A|B,C) = P(A|C)$$
If the outcome of B doesn't have an effect on the outcome of $A$, then why would the outcome of $B$ AND $C$ have an effect on $A$ that is different than the effect of the outcome of just $C$?
Is there perhaps a simple counterexample to illustrate this? For example, let's say $A$ and $B$ are the event of picking an ace of spades from a deck of cards (each has their own deck of cards).

My favourite example is a chessboard: if you pick a point uniformly then the row, column, and colour are pairwise independent.
Suppose A is colour, B is row, and C is column. Then P(A="white"|B)=0.5 and P(A="white"|C)=0.5 for any value of B, but P(A="white"|B,C) is either 0 or 1 for any values of B and C.
