Suppose I have a statement $p(m,n)$ where $m,n \in \mathbb{N}$ that I want to prove.
Suppose I have proofs of the following:
$p(1,n)$ is true for all $n \in \mathbb{N}$.
$p(m,1)$ is true for all $m \in \mathbb{N}$.
If $p(m-1,n)$ and $p(m,n-1)$ are true for some $m,n \in \mathbb{N}$, then $p(m,n)$ is true; i.e. I have proof for $p(m,n-1) \land p(m-1,n) \implies p(m,n)$.
Now my question is: How do I prove that if I have those 3 proofs then the statement $p(m,n)$ is true for all $m,n \in \mathbb{N}$?
Suppose the statement is false for some $m$ and some $n$. Then there is a least a least $m^{\ast}$ for which there is an $n$ such that the statement is false for $m^{\ast}$ and $n$. But now there is a least $n^{\ast}$ such that the statement is false for $m^{\ast}$ and $n^{\ast}$ By hypothesis 1 and 2 we know that both $m^{\ast}$ and $n^{\ast}$ are greater than $1$. Now apply hypothesis 3.