Is the following guaranteed to be true for any covariant vector $f_\mu$ (1-form $\boldsymbol{f}$) in the absence of torsion? $$\nabla_{[\alpha}\nabla_{\beta}f_{\mu]}=\partial_{[\alpha}\partial_{\beta}f_{\mu]}=\boldsymbol{d}\boldsymbol{d}\boldsymbol{f}=0,$$ where $\nabla_{\alpha}$ is the covariant derivative, $\partial_{\beta}$ is the partial derivative, and $\boldsymbol{d}$ is the exterior derivative, and brackets in the subscript means antisymmetrization.
$\partial_{[\alpha}\partial_{\beta}f_{\mu]}=\boldsymbol{d}\boldsymbol{d}\boldsymbol{f}$ is just the definition of $\boldsymbol{d}$ and is everywhere in textbooks, and the fact that it is zero is also all over the place. So my real question is:
Is the following always true?
$\nabla_{[\alpha}\nabla_{\beta}f_{\mu]}=\partial_{[\alpha}\partial_{\beta}f_{\mu]}$