Suppose we have two simple closed curves $\alpha$ and $\beta$ on an $n$-times punctured disc $D_n$ which bound the same boundary components. Is it true that there is a $\phi\in MCG(D_n)$(mapping class group of $D_n$) such that $\phi(\alpha)=\beta$? If yes, why is it true? and is the element $\phi$ unique in any sense?
PS: We say a curve $C$ on $D_n$ bounds certain boundary components if they are all inside one of the two regions created by $C$ on $D_n$.