Consider the following result from General relativity:
$$(1) \space\space\Gamma^{j}_{ij} = \frac{1}{2} g^{jk} (\partial_i g_{jk} + \partial_j g_{ki} - \partial_{k}g_{ij} ) $$
My question is whether it is a general principle in tensor calculus when we have repeated symbols in the same term that these symbols become arbitrary and can be replaced with any other symbol. For instance if we expand (1) as below
$$ (2) \space \space\Gamma^{j}_{ij} = \frac{1}{2} g^{jk} \partial_i g_{jk} + \frac{1}{2} g^{jk} \partial_j g_{ki} - \frac{1}{2} g^{jk}\partial_{k}g_{ij} $$
then we can just replace all jk indices however we wish? In doing so then would it be appropriate to write (2) as (3)
$$ (3) \space \space\Gamma^{j}_{ij} = \frac{1}{2} g^{jk} \partial_i g_{jk}$$
since the metric tensor $g_{ij} = g_{ji} $ due to its symmetrical nature.
You are correct. More precisely, you also need the symmetry of $g^{jk}$. $$\begin{align*} g^{jk}\partial_jg_{ki} &= g^{jk}\partial_jg_{ik} & \text{since $g_{ki}=g_{ik}$}\\ &= g^{kj}\partial_kg_{ij} & \text{renaming $j$ and $k$}\\ &= g^{jk}\partial_kg_{ij} & \text{since $g^{kj}=g^{jk}$} \end{align*}$$ so indeed the second and third terms cancel out.