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.

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 given description of the weak equivalences without first proving the comparison theorem.