Two random variables $A$ and $B$ are conditionally independent (when conditioned on a set of random variables $\mathcal{C}$), if the variables are d-separated, i.e. if all the paths from $A$ to $B$ are blocked.
Suppose we have a graph like this.
$ (A) \rightarrow (C) \leftarrow (B) $
We want to evaluate if $A \perp\mkern-10mu\perp B$. We immediately see that $C$ is a collider, but it's not in the conditioning set (the conditioning set is empty), so $C$ blocks the path between $A$ and $B$. There is only one path between $A$ and $B$, and it's blocked, so $A$ and $B$ are d-separated by $C$.
D-separation between $A$ and $B$ implies $A \perp\mkern-10mu\perp B\, |\, C$ to be true. So let's check if $A \perp\mkern-10mu\perp B\, |\, C$ is true. It's clearly false because $C$ is in the conditioning set, $A$ and $B$ are not blocked nor d-separated.
Am I missing something? Why do I get such a paradoxical result?
The d-separation of nodes in the unconditioned DAG does not imply that the nodes will be d-separated when conditioned. A collider in a conditioning set does not block the path.
In the DAG $(A)\to(C)\gets(B)$, the collider $(C)$ blocks the path between nodes $(A)$ and $(B)$. $(A),(B)$ are d-separated. $$A\perp\!\!\!\!\perp B$$
When $(C)$ is a member of the conditioning set, the path is unblocked. $(A),(B)$ are d-connected when conditioned by $\{C\}$. $$A\not\perp\!\!\!\!\!\perp B\mid C$$
If a node in a connecting path is in the conditioning set, the node becomes a blocker.
If a node in a conditioning set is a collider, or a decendant of a collider, the collision is not a blocker.