If $mx=m$ for some (or all) integer $m$, then $x=1$

Question

I am studying proofs, and I am stuck thinking about the logic behind these two propositions:

1. Let $x \in\mathbb Z$, if $x$ has the property the for all $m \in\mathbb Z$ $mx = m$, then $x=1$.

2. Let $x \in\mathbb Z$, if $x$ has the property the for some nonzero $m \in\mathbb Z$ $mx = m$, then $x=1$.

Here is a solution:

\begin{align*} mx &= m \\ mx &= m \cdot 1 \\ x &= 1\\ \end{align*}

I can get the same solution for both propositions but the difference seems to be at the third step. For 1) I can simply simplified both m. 2) I can use the cancellation axion: Let $m,n$ and $p$ be integers. If $m \cdot n = m \cdot p$ and $m \ne 0$, then $n = p$.

However, I could be wrong and would appreciate the community's feedback. Thank you!