# Negating a statement with “=” in it.

I am trying to negate this following statement: $(\forall y)(\exists x)[y = f(x)]$ I negated the statement but stopped at the $\neg[y = f(x)]$ part. I wasn't to sure how to negate it when it came to an equal sign, my thinking would have been replacing "=" with "Not equals($\neq$)", is it okay to leave the negation of $\neg[y = f(x)]$ the way it is or that's a big NO NO? Here is my progressions so far: $$= (\exists y)\neg\Big[(\exists x)[y = f(x)]\Big]$$
$$= (\exists y)(\forall x)\neg[y = f(x)]$$

Yes, it's okay to use $\neq {}$.
• @KonoDioDa Note, $x \neq y$ is usually by definition $\neg(x = y)$ in which case you aren't actually doing anything. In other words, $x = y$ is treated just like any other atomic predicate $P(x,y)$ which is to say there is no rule (or need) to "simplify" its negation. – Derek Elkins Sep 11 '17 at 21:09