While Dylan's comment is correct, it doesn't really explain how this is shown.  It's actually a corollary of the existence of the Joyal model structure, which is proven earlier in the chapter as a corollary of the comparison theorem with simplicial categories.

Since the cofibrations for both the Joyal and Kan-Quillen model structures coincide, and since the fibrant objects for the Kan-Quillen model structure are also fibrant for the Joyal model structure, it follows that the Kan-Quillen model structure is a left-Bousfield localization of the Joyal model structure.

It is a general fact of the theory of (left) Bousfield localization (see e.g. Hirschhorn) that the local equivalences between local objects are exactly the equivalences in the original unlocalized model structure.

I don't see any way to show this directly just from the description of the weak equivalences without first proving the comparison theorem.