Negating a statement with "=" in it.

Question

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: $$ = (\exists y)\neg\Big[(\exists x)[y = f(x)]\Big]$$
$$ = (\exists y)(\forall x)\neg[y = f(x)]$$

Answer 1

Yes, it's okay to use $\neq {}$.

Answer 2

Why didn't you continue that way you had been thinking? It would have been correct.