Wilson's theorem could be simplified?

Question

I know that Wilson's theorem states that a number n is prime iff

   (n - 1)! congruent to -1 (mod n)

However, since

   (n - 1) congruent to -1 (mod n)

Why isn't the theorem simplified to the following condition?

  (n - 2)! congruent to 1 (mod n)

Answer 1

It is indeed true that $n\ge 2$ is prime if and only if $(n-2)!\equiv 1\pmod n$.

"Why isn't the theorem simplified to this condition?": why should it? It looks basically the same to me.