This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.
Start with the Bianchi identity
Contract both sides of the above equation with a pair of metric tensors:
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
The last two terms are the same (changing dummy index n to m) and can be combined into a single term which shall be moved to the right,
which is the same as
Swapping the index labels l and m yields
The last equation in Proof 1 above can be expressed as
where δ is the Kronecker delta. Since the mixed Kronecker delta is equivalent to the mixed metric tensor,
and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then
Factor out the covariant derivative
then raise the index m throughout
The expression in parentheses is the Einstein tensor, so