Question about the Dirac adjoint and Feynman slash notation

I was trying to prove the identity $overline{displaystyle{not}{a}displaystyle{not}{b}dots displaystyle{not}{p}} = displaystyle{not}{p}dots displaystyle{not}{b}displaystyle{not}{a}$. On simplifying the LHS I end up with $ overline{ displaystyle{not}{p}^{}} dots overline{ displaystyle{not}{b}^{}} overline{ displaystyle{not}{a}^{}} $.

I’m wondering if $ overline{ displaystyle{not} p} = gamma^0 displaystyle{not} p^{dagger} gamma^0 = displaystyle{not} p$ as this would let me complete the final step.