Alpha conversion renaming

Example:

$$(\lambda x.\ \lambda y.\ x)\ y \to [x \to y] \lambda y.\ x$$

Without alpha-conversion, performing a naive substitution would result in:

$$[x \to y]\lambda y.\ x = \lambda y.\ \ y$$

So, a function that was always returning a constant $x$ becomes an identity function.

With alpha-conversion, we can correctly perform a substitution:

$$[x \to y] \lambda y.\ x = [x \to y]\lambda z.\ x = \lambda z.\ y$$