Formal proof that if $a-b = \frac{a-b}{ab}$, $a-b$ has to be zero.

Question

Just out of interest, how would I provide a proof that if $a-b = \frac{a-b}{ab}$, then $a$ has to be equal to $b$? It appears really logical, I just want to know how to formally prove it. Thanks for any suggestions.

Answer 1

Per lulu's comment, it's not true, but this is true:

$a-b=\dfrac{a-b}{ab}\implies (a-b)ab=a-b \implies (a-b)(ab-1)=0\implies a=b$ or $ab=1$