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).

As @Sextus Empiricus (the ancient skeptic!) said, collider bias phenomenon in epidemiology is a great exemplar of why marginal independence of two random variables does not necessarily imply their conditional independence, given a third random variable.
Sackett (1978)'s example is a classic for the collider bias.  In his paper, $A =$ having locomotor disease, and $B = $ having respiratory disease. Let $C$ denote the event that a person is hospitalized (for whatever reason). He observed that among hospitalized patients, there was a strong association between $A$ and $B$, whereas there was no association between $A$ and $B$ in the overall population (mix of both $C$ and $\bar C$).  If $C$ denotes hospitalization, then we see that $P(A|B,C) \ne P(A|C)$ and $P(B|A,C) \ne P(B|C)$, even though $P(A|B) = P(B)$, this is because both $A$ and $B$ increase the probability of $C$.   Hospitalization, $C$, is the collider, being a common effect of $A$ and $B$.
