Calculate curvature by coordinate component method

PhyPsy

I'm trying to follow the math in Wald's General Relativity where he starts out with the equation for covariant derivative:
[itex]\nabla[/itex]_b[itex]\omega[/itex]_c = [itex]\partial[/itex]_b[itex]\omega[/itex]_c - [itex]\Gamma[/itex]^d_bc[itex]\omega[/itex]_d

He uses that to derive the equation for a double covariant derivative:
[itex]\nabla[/itex]_a[itex]\nabla[/itex]_b[itex]\omega[/itex]_c = [itex]\partial[/itex]_a([itex]\partial[/itex]_b[itex]\omega[/itex]_c - [itex]\Gamma[/itex]^d_bc[itex]\omega[/itex]_d) - [itex]\Gamma[/itex]^e_ab([itex]\partial[/itex]_e[itex]\omega[/itex]_c - [itex]\Gamma[/itex]^d_ec[itex]\omega[/itex]_d) - [itex]\Gamma[/itex]^e_ac([itex]\partial[/itex]_b[itex]\omega[/itex]_e - [itex]\Gamma[/itex]^d_be[itex]\omega[/itex]_d)

Now, using the Riemann tensor definition R_abc^d[itex]\omega[/itex]_d = [itex]\nabla[/itex]_a[itex]\nabla[/itex]_b[itex]\omega[/itex]_c - [itex]\nabla[/itex]_b[itex]\nabla[/itex]_a[itex]\omega[/itex]_c, this equation is derived:
R_abc^d[itex]\omega[/itex]_d = [itex]\partial[/itex]_[a[itex]\partial[/itex]_b][itex]\omega[/itex]_c - [itex]\omega[/itex]_d[itex]\partial[/itex]_[a[itex]\Gamma[/itex]^d_b]c - [itex]\Gamma[/itex]^d_c[b[itex]\partial[/itex]_a][itex]\omega[/itex]_d - [itex]\Gamma[/itex]^e_[ab]([itex]\partial[/itex]_e[itex]\omega[/itex]_c - [itex]\Gamma[/itex]^d_ec[itex]\omega[/itex]_d) - [itex]\Gamma[/itex]^e_c[a[itex]\partial[/itex]_b][itex]\omega[/itex]_e - [itex]\Gamma[/itex]^e_c[a[itex]\Gamma[/itex]^d_b]e[itex]\omega[/itex]_d

I know that the term [itex]\partial[/itex]_[a[itex]\partial[/itex]_b][itex]\omega[/itex]_c cancels, and, due to the symmetry of the metric connection, [itex]\Gamma[/itex]^e_[ab]([itex]\partial[/itex]_e[itex]\omega[/itex]_c - [itex]\Gamma[/itex]^d_ec[itex]\omega[/itex]_d) also cancels, but the next step in the book also has a couple other terms cancelled out:
[itex]\Gamma[/itex]^d_c[b[itex]\partial[/itex]_a][itex]\omega[/itex]_d, and
[itex]\Gamma[/itex]^e_c[a[itex]\partial[/itex]_b][itex]\omega[/itex]_e

I don't see how these terms cancel. Can someone help me?