I need your help in order to prove this relation:

$$\gamma^{\mu} \not p \gamma_{\mu}$$ that has to give me $$-2 \not p$$

I tried this: $$ \begin{array}{rll} \gamma^{\mu} \not p \gamma_{\mu} & = \gamma^{\mu}\big(\gamma^{\nu}p_{\nu}\big)\gamma_{\mu} \\ & = \gamma^{\mu}\big(\gamma^{\nu}\gamma_{\mu}\big)p_{\nu} \\ & = \gamma^{\mu}\big(\gamma^{\nu}\gamma_{\mu} + \gamma^{\nu}\gamma_{\mu} - \gamma^{\nu}\gamma_{\mu}\big)p_{\nu} \\ & = \gamma^{\mu}\big(2\eta^{\nu}_{\mu} - \gamma^{\nu}\gamma_{\mu}\big)p_{\nu} \\ & = 2\gamma^{\mu}\eta^{\nu}_{\mu} - \gamma^{\mu}\gamma^{\nu}\gamma_{\mu}p_{\nu} \\ & = 2\gamma^{\nu}p_{\nu} - 4\gamma^{\nu}p_{\nu} \\ & = -2\not p \end{array} $$

My question is: is this valid? And in the second line: is the passage valid? Can I exchange the order of $\gamma$ and $p$?

Thank you everybody!!