One of the axioms out there is: if n and m are integers and they have the same next integer then n=m
But why is it considered to be an axiom since I can easily prove it:
Have the same next integer means: n+1=m+1 <=> n=m
If you leave out this axiom you can satisfy Peano's other axioms with a set that is not the natural numbers.
Consider the set containing $0,a_1,a_2$ with $0\neq a_1,a_2$.
We can define
This set satisfies all of Peano's axioms except that in this case successor of two numbers being equal does not imply that the numbers are equal ($0$ and $a_2$ have the same successor).
The axiom is then in some sense really a definition. If you leave it out you get a theory of arithmetic that is different from the usual one. This could be compared to leaving out the parallel line postulate to get non-Euclidean geometries.
The original axiom is part of a very small set of axioms needed to derive the usual properties of addition and multiplication. Instead of that one axiom you took all the usual properties of addition as given.
There are multiple sets of axioms that can be used to define the natural numbers. We have largely settled on the Peano axioms as the simplest definition of the natural numbers. Along with a formal definition of equality, Peano's axioms define the natural numbers as a set $ N $ with a function $ S: N \rightarrow N $ and an element $ 0 \in N $ such that:
Given these axioms, we can define addition recursively as:
and define multiplication as:
Given this foundation, we can prove that addition and multiplication are commutative and associative and that multiplication distributes over addition. That is left as an exercise for the reader.
To address the questioner's proof, the proof was stated like this:
However, this assumes that the principle trying to prove is true. Therefore the proof is begging the question.