Dirac equation for bilinear Lorentz scalar

Question

Let $O(x)$ denote a bilinear field operator, such that

$$\begin{equation}\label{eq:eq1} O(x) = \bar{\psi}(x)\psi(x) \end{equation}$$

where $\bar{\psi} \equiv \psi^\dagger \gamma^0$. I'm supposed to prove that

$$\begin{equation}\label{eq:eq2} (\partial^2-m^2) \bar{\psi}(x)\psi(x) = 0 \end{equation}$$

**My attempt**

I did the following:

$$\begin{equation} (\partial^2-m^2) \bar{\psi}(x)\psi(x) = 0 \\ \Leftrightarrow (\partial^2 \bar{\psi})\psi+\bar{\psi}\partial^2\psi-m^2\bar{\psi}\psi = 0  \end{equation}$$

Using the slash notation, 

$$\begin{equation} i \displaystyle{\not} \partial(-i \displaystyle{\not}\partial\bar{\psi})\psi+\bar{\psi}(-i\displaystyle{\not}\partial)i\displaystyle{\not}\partial\psi-m^2\bar{\psi}\psi = 0 \\ \Leftrightarrow i \displaystyle{\not} \partial(m\bar{\psi})\psi+\bar{\psi}(-i\displaystyle{\not}\partial)m\psi-m^2\bar{\psi}\psi = 0 \\ \Leftrightarrow -m^2\bar{\psi}\psi-m^2\bar{\psi}\psi-m^2\bar{\psi}\psi = 0 \\ \Leftrightarrow -3m^2\bar{\psi}\psi = 0 \end{equation}$$

which would imply that $m = 0$. What am I doing wrong?