  We give a new proof of the following theorem: moduli spaces of stable
complexes on a complex projective K3 surface, with primitive Mukai vector and
with respect to a generic Bridgeland stability condition, are hyperk\"{a}hler
varieties of $\mathrm{K3}^{[n]}$-type of expected dimension. We use derived
equivalences, deformations and wall-crossing for Bridgeland stability to reduce
to the case of the Hilbert scheme of points.
