Klein-Gordon Green's Function

I am having trouble understanding the derivation presented in chapter 2 of Overview of Quantum Field Theory, in which the authors show that the a particular function, denoted as $D_R(x-y)$ is the Green’s function for the Klein-Gordon equation. Here is the derivation for $D_R(x – y)$, which is equation (2.54) in the book:

$langle0|[phi(x), phi(y)] |0rangle = int frac{d^3p}{(2pi)^3}frac{1}{2E_p}big(e^{-ipcdot (x – y)} – e^{ipcdot (x – y)} big) = int frac{d^3p}{(2pi)^3}int frac{dp^0}{2pi i}frac{-1}{p^2 – m^2}e^{-ipcdot (x – y)}$

$D_R(x – y) equiv theta(x^0 – y^0)langle0|[phi(x), phi(y)] |0rangle$

The results of equation 2.56 (see below) are confusing me. I can derive the equations after the first equals sign (it is just the product rule); however, right after the second equals sign I see that the first term contains a $pi(x)$. Why can we say this, because after all the derivative on the first term applies only to the step function $theta(x^0 – y^0)$? Also after the first equal sign, I am confused again: why is the expression after the second equals sign equal to the four dimensional Dirac delta? I believe it has something to do with equation 2.54 but I am not sure how the math works; can someone explain?

$(partial^2 + m^2)D_R(x – y) = (partial^2 theta(x^0 – y^0))langle0|[phi(x), phi(y)] |0rangle + 2(partial_mu theta(x^0 – y^0))(partial^{mu}langle0|[phi(x), phi(y)] |0rangle) + theta(x^0 – y^0)(partial^2 + m^2)langle0|[phi(x), phi(y)] |0rangle$

$= -delta(x^0 – y^0)langle0|[pi(x), phi(y)] |0rangle + 2delta(x^0 – y^0)langle0|[pi(x), phi(y)] |0rangle + 0$

$= – idelta^4(x – y)$