Prove that Choice Coherence Implies IIA
I think your notation makes this more confusing that is necessary since $A$ and $B$ represent sets of alternatives in one setting and alternatives in the other setting.
So write IIA as:
If $x\in C\big(\{x,y\}\big)$ and $y\notin C\big(\{x,y\}\big)$, then $y\notin C\big(x,y,z\big)$.
To prove this from choice coherence, simply choose $A$ and $B$ appropriately.