Is the Inner Product for Dirac Spinors Antisymmetric?

[LAHLH]

Homework Statement

Show that \psi (\gamma^a\phi)=-(\gamma^a\phi)\psi

Homework Equations

Maybe \{\gamma^a, \gamma^b\}=\gamma^a\gamma^b+\gamma^b\gamma^a=2\eta^{ab}I

Perhaps also:

(\gamma^0)^{\dag}=\gamma^0 and (\gamma^i)^{\dag}=-(\gamma^i)

The Attempt at a Solution

The gammas are matrices so I guess we start with

\psi_{\mu}[(\gamma^a)^{\mu\nu}\phi_{\nu}]
=\psi_{\mu}[(((\gamma^a)^*)^{\dag})^{\nu\mu}\phi_{\nu}]
=-[(((\gamma^a)^*))^{\nu\mu}\psi_{\mu}]\phi_{\nu}

Which looks almost correct except the *, and also I'm not sure if I was supposed to assume that a can only refer to spatial indices, not the 0 which is equal to its hermitian conj, not minus it.

Thanks for any help

 

[LAHLH]

Anyone?