Is there a name for this method of proof?

Question

If I want to prove that some statement $P(n,m)$ is true for all $n, m \in \mathbb{N}$, I can do this by showing that:

1. $P(1, 1)$
2. $\forall n \in \mathbb{N}: P(1, n) \implies P(1, n+1)$
3. $\forall n, m \in \mathbb{N}: P(n, m) \implies P(n+1, m)$

Is there a name for this type of proof?

Answer 1

It is called induction.

Or you may call it two nested inductions: With (1) and (2), you show that $P(1,m)$ holds for all $m\in \Bbb N$, by induction on $m$. Then with this and $(3)$, you show by induction on $n$ that $P(n,m)$ holds for all $n$ (where $m\in\Bbb N$ is arbitrary!)