Surgery on links

Question

Suppose $K^m \subset Y^n$ is a link of spheres $S^m_1, \cdots, S^m_i$ with trivialized normal bundle in a smooth manifold $Y^n$. I can do surgery on $Y^n$ using $K^m$. Is it true that the resulting manifold $Y'$ is independent of the order in which I do surgery on $S^m_1, \cdots, S^m_i$? That is, if I do surgery on $S^m_1$ and then on $S^m_2$, is this the same as doing surgery first on $S^m_2$ and then on $S^m_1$?