covariant derivatives and christoffel symbols

$\displaystyle 0=\frac{\partial{g_{mn}}}{\partial{y^p}}+ \Gamma_{pm}^sg_{sn} + \Gamma_{pn}^rg_{mr}$

i am trying to verify this equality so here is what i did so far:

$\displaystyle \frac{\partial{g_{mn}}}{\partial{y^p}}+\frac{1}{2} g^{sd}(\frac{\partial{g_{dm}}}{\partial{y^p}}+\fra c{\partial{g_{dp}}}{\partial{y^m}}-\frac{\partial{g_{pm}}}{\partial{y^d}})g_{sn}+\fra c{1}{2}g^{rz}(\frac{\partial{g_{zn}}}{\partial{y^p }}+\frac{\partial{g_{zp}}}{\partial{y^n}}-\frac{\partial{g_{pn}}}{\partial{y^z}})g_{mr}$

then since $\displaystyle g^{sd}g_{sn}=\delta^d_n$ i now have

$\displaystyle \frac{\partial{g_{mn}}}{\partial{y^p}}+\frac{1}{2} \frac{\partial{g_{nm}}}{\partial{y^p}}+\frac{1}{2} \frac{\partial{g_{np}}}{\partial{y^m}}-\frac{1}{2}\frac{\partial{g_{pm}}}{\partial{y^n}}+ \frac{1}{2}\frac{\partial{g_{mn}}}{\partial{y^p}}+ \frac{1}{2}\frac{\partial{g_{mp}}}{\partial{y^n}}-\frac{1}{2}\frac{\partial{g_{pn}}}{\partial{y^m}}$

and i seem to be stuck here. have what i've been doing so far correct and if so how may i continue. thanks.