I'm studying some complex analysis in preparation for a qualifier exam and I'm doing exercise $6.12$ from Robert Greene and Steven Krantz' book Function Theory of One Complex Variable.
I have $\Omega$ a simply connected domain in $\mathbb{C}$, with $P,Q \in \Omega$ two different points. $\phi_1 : \Omega \to \Omega$ and $\phi_2 : \Omega \to \Omega$ are conformal maps (that is, bijective and holomorphic) such that $\phi_1(P) = \phi_2(P)$ and $\phi_1(Q) = \phi_2(Q)$. Then the question is to prove that $\phi_1 \equiv \phi_2$.
I don't know why but I've been thinking about how to approach this without success for quite some time now. I've thought of maybe using the Riemann mapping theorem to say that there's a conformal map $\phi:\Omega \to \mathbb{D}$ to the open unit disk, and similar things but I'm not getting anywhere really.
Thus I would really appreciate some help with this problem, maybe some hints on how to proceed would be very appreciated. Thanks in advance for any help.