About the proof of the second Bianchi Identity

The second Bianchi Identity is $$\nabla_{[a}R_{bc]de}=0$$

As far as I know, the proof (say, Wolfram MathWorld) starts by stating the representation of Riemann tensor in local inertial coordinates $$R_{abcd}=\frac{1}{2}(\partial_a\partial_cg_{bd}-\partial_a\partial_dg_{bc}-\partial_b\partial_cg_{ad}+\partial_b\partial_dg_{ac}).$$

Then we calculate $$\partial_aR_{bcde}$$

accordingly, and we say that it's true in a local inertial coordinate, and after changing the partial derivative into a covariant derivative, it's true in general.

My concern is, that I think we cannot express the Riemann tensor and the covariant derivative into local frames one by one, but should simultaneously. Say $$\nabla_{a}R_{bcde}=\frac{1}{2}(\partial_a+\Gamma_1)(\partial_a\partial_cg_{bd}-\partial_a\partial_dg_{bc}-\partial_b\partial_cg_{ad}+\partial_b\partial_dg_{ac}+\Gamma_2)$$

where $$\Gamma_1$$ and $$\Gamma_2$$ are some terms involving the Christoffel symbol. When we only concern $$R_{bcde}$$ in a local frame, $$\Gamma_2$$ vanishes. But now we have a new term $$\partial_a\Gamma_2$$

which I cannot see vanish because it involves a derivative of the Christoffel symbol. So I think in a local frame $$\nabla_aR_{bcde}$$ is not $$\partial_aR_{bcde}$$.

Is there anything wrong?