given a $\lambda \text{-}$term $M$, how does one prove that the identity substitution on $M$ results in $M$ ? That is:$$[x/x] M \equiv M$$
Should I go for the cases, trying to prove for $M$ atomic, for $M = \lambda x.N$ and for $M = AB\;$?
If not, what approach should I use to prove that?