Confusion on Bianchi Identity proof

[Dazed&Confused]

This is from a general relativity book but I think this is the appropriate location.

The proof that $$\nabla_{[a} {R_{bc]d}}^e=0$$
is as follows:

Choose coordinates such that $\Gamma^a_{bc}=0$ at an event. We have $$\nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a \partial_c \Gamma^e_{bd} + \textrm{ terms in } \Gamma \partial \Gamma \textrm{ and } \Gamma \Gamma \Gamma.$$

Because the first term on the right-hand side is symmetric in $ab$ and the second in $ac$, and because the other terms vanish at the event, we have $$\nabla_{[a}{R_{bc]d}}^e=0$$
at the event in this coordinate system. However, this is a tensor equation, so it is valid in every coordinate system.

The bit I'm unsure of is how ''the other terms'' vanish at the event, so help would be appreciated.

 

[Orodruin]

"The other" terms have the Christoffel symbols as a factor and the Christoffel symbols are all zero because of the choice of coordinates. (You can always find such coordinates for any given event.)

 

[Dazed&Confused]

Thanks I think I had a mental block because of the index notation.