quantum mechanics - Can I replace eigenvalue of $p$ operator with position space representation of $p$ operator?

$$\begin{equation} \begin{aligned} \langle x|\hat{p}|\psi\rangle &= \int dp\ \langle x|\hat{p}|p\rangle \langle p|\psi\rangle\\ &=\int dp\ p\langle x|p\rangle \langle p|\psi\rangle \\ &=\int dp \ \left(-i\hbar \frac{\partial}{\partial x}\right) \langle x|p\rangle \langle p|\psi\rangle \end{aligned} \end{equation}$$

Please explain how we can go from second step to third step? In the second step, the $p$ is an eigenvalue which has been replaced by position representation of the momentum operator in the third step. How can we replace a number by an operator?