Proof of $\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

Question

I'm a student of physics. There is an identity in tensor calculus involving Kronecker deltas ans Levi-Civita pseudo tensors is given by $$\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ which is extensively used in physics in deriving various identities. I have neither found a proof of this in physics textbooks nor in Wikipedia. In particular, how does the above formula follow from the definition of $\epsilon_{ijk}$ tensor$$\epsilon_{ijk} = \begin{cases} +1 & \text{ for even permutations }, \\ -1 & \text{ for odd permutations } ,\\ \;\;\,0 & \text{ for repetition of indices }, \end{cases}$$ This is the only definition of I'm familiar with.