$R$ squared gravity reduced field equation

Given is the static and spherically symmetric metric ($c=1$)
begin{equation}
ds^2 = g^mathrm{(0)}_mathrm{munu}dx^mu dx^nu = -e^mathrm{2nu(r)}dt^2+e^mathrm{2lambda(r)}dr^2-r^2(dtheta^2+sin^2(theta)dvarphi^2).
end{equation}
The Einstein field equations are
begin{equation}
G_mathrm{munu} = 2partial_muphipartial_nuphi-g_mathrm{munu}partial^gammaphipartial_gammaphi+8pi T_mathrm{munu}-frac{1}{2}V(phi)g_mathrm{munu}
end{equation}
where $G_mathrm{munu}$ is the known Einstein tensor and $phi$ is a scalar field, for which is the scalar field equation
begin{equation}
nabla_munabla^muphi = 4pifrac{1}{sqrt{3}}g^mathrm{munu}T_mathrm{munu}+frac{1}{4}frac{dV}{dphi}.
end{equation}
$T_mathrm{munu} = (rho+p)u_mu u_nu+pg_mathrm{munu}$ is the stress-energy tensor for a perfect fluid, $rho$ is the rest mass density and $p$ the pressure. Now we go in to the local rest frame and therefore $u_mu = (e^nu,0,0,0)$ and we assume that the scalar field depends only on the radial coordinate $phi=phi_0(r)$.

The form of the metric $ds^2$ is like the Schwarzschild metric and therefore the Ricci tensor and hence the Einstein tensor only have the four diagonal elements.

Now I want to derive the reduced field equations fore this specific problem. There are three Einstein equations ($G_mathrm{varphivarphi}$ depends liearly on $G_mathrm{thetatheta}$) and one for the scalar field. The solutions are in a paper (Yazadjiev et al JCAP06(2014)003) equation (2.17) up to (2.20)
begin{align}
&(1)quad frac{1}{r^2}frac{d}{dr}[r(1-e^{-2lambda})] = 8pirho+e^{-2lambda}(frac{dphi_0}{dr})^2+frac{1}{2}V(phi_0)
&(2)quad frac{2}{r}e^{-2lambda}frac{dnu}{dr}(1-e^{-2lambda}) = 8pi p+e^{-2lambda}(frac{dphi_0}{dr})^2-frac{1}{2}V(phi_0)
&(3)quad frac{dp}{dr} = -(rho+p)left(frac{dnu}{dr}-frac{1}{sqrt{3}}frac{dphi_0}{dr}right)
&(4)quad frac{d^2phi_0}{dr^2}+left(frac{dnu}{dr}-2frac{dlambda}{dr}+frac{2}{r}right)frac{dphi_0}{dr} = frac{4pi}{sqrt{3}}(3p-rho)e^{2lambda}+frac{1}{4}frac{dV}{dphi_0}e^{2lambda}.
end{align}
So my problem is the third equation. The first one comes from the Einstein equation with $mu=nu=0$, the second one comes from $mu=nu=r$ and the fourth comes from the scalar field equation. I do not know how to derive the third equation. I calculate the Einstein equation with $mu=nu=theta$ but it’s not equal to the third one. Maybe it’s a superposition with some Einstein equations and the scalar field equation or I need to calculate the bianchi identity? Does someone have a hint for me?