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).

Imagine $A$ and $B$ are independent Bernoulli random variables with success probability 1/2, and let $C = AB + (1 - A)(1 - B)$ (if you like, $A$ is the event that a coin lands heads, $B$ is the event that another coin lands heads, and $C$ is the event that both coins land the same side up). Then $P(A = 1 \mid C = 1) = 1/2$ but $P(A = 1 \mid B = 1, C = 1) = 1$.
A bit more generally, a common situation where this is not true is when $(A, B, C)$ form a v-structure, i.e., where $A \rightarrow C \leftarrow B$, where $A$ and $B$ are marginally independent but where $B$ can provide information about $A$ if both cause $C$.
