# Levi-Civita Symbol and index manipulation

Above is the question and answer to question (b) (ignore (a))... I don't get where the final implication comes from. Why can we use c as a (covariant) index on the LHS... surely we must use d or something different to c. Is the implication obvious, or do we have to justify all the possible combinations in order to understand the implication?

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Since each side is vector valued, It doesn't matter what you call the index. For instance, say you called one $c$ and the second $d$, you still would have that: $$LHS_{c1}=LHS_{d1},~ LHS_{c2}=LHS_{d2}, ~LHS_{c3}=LHS_{d3}$$ So it these are essentially the same.
The only thing that happens in the last impliciation is that the left-hand side is rewritten using $\mathbf e_a\times \mathbf e_b = \epsilon_{abc}\mathbf e_c$ (which is close to being a definition of the cross product).