A friend of mine showed me this proof to demonstrate that any algebra variable does not equal any number.
his proof relies on this idea $$0\neq1$$ $$0x\neq1x$$ $$0\neq1x$$ $$0\neq x$$ Full Proof: $$\forall n (n \in \Re)$$ $$0\neq1\Rightarrow0(x-n)\neq1(x-n)$$ $$0(x-n)\neq1(x-n)\Rightarrow0\neq x-n$$ $$0\neq x-n\Rightarrow n\neq x$$
I feel like it is a divide by zero fallacy but I'm yet to pinpoint the exact error
The false implication $x \neq y \to ax \neq ay$ is used. This is true iff $a$ is non-zero (in a field), because then the equivalent (by contraposition) $ax = ay \to x=y$ can be shown by division by $a$. So indeed it's a division by $0$ error implicitly.
The first line in your extended (non)proof is wrong when $x=n$. So it's not true for all $x$ and $n$.
You can't multiply inequations like this: $a\neq b$ does not imply $ac\neq bc$. Indeed, if $c=0$, then $ac=0=bc$ is always true, even if $a\neq b$.
(If you know that $c\neq 0$, then this step would be valid, since if $ac$ were equal to $bc$ you could divide both by $c$ to get $a=b$ which contradicts the fact that $a\neq b$. So, in a sense, this is indeed a division by $0$ error.)
When we multiply for $(x-n)$ we need to set $x\neq n$ that is precisely the solution we obtain.
