I am reading a book where the following proposition is proved by induction:
If $n$ is a natural number such that $n>1$, then $n-1$ is a natural number.
Now, since the usual way of proving a statement of the form $\forall n \in \mathbb{N} \; \; P(n)$ is to show that $\{ n \in \mathbb{N} \mid P(n) \} = \mathbb{N} $, a possible approach would be to show that
$\{ n \in \mathbb{N} \mid \text{If} \; n>1, \; \text{then} \; n-1 \in \mathbb{N} \} = \mathbb{N} $ by induction.
However, the textbook instead shows that
$\{ n-1 \in \mathbb{N} \mid n \in \mathbb{N} \;\text{and} \; n>1 \} = \mathbb{N} $
My questions is : Why is this last statement equivalent to the first proposition?