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?