Identity with covariant derivatives and curvature tensor

Question

Let $\nabla_a$ be a covariant derivative, $R^a_{~~bcd}$ the curvature tensor and $w_a$ a one form. The following identity is well known (in fact I think some even define the curvature tensor through this); $$\big[\nabla_a,\nabla_b\big]w_c=-R^d_{~~cab}w_d.$$ In this identity, may I replace $w_c$ with $\nabla_c$ and conclude that $\big[\nabla_a,\nabla_b\big]\nabla_c=-R^d_{~~cab}\nabla_d$, or is this too naive?