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

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

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    $\begingroup$ Ah okay, thanks. $\endgroup$ – Kono Dio Da Sep 11 '17 at 20:29
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    $\begingroup$ @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. $\endgroup$ – Derek Elkins Sep 11 '17 at 21:09
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Why didn't you continue that way you had been thinking? It would have been correct.

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  • $\begingroup$ I thought it would have be more complex than that haha. Thanks. $\endgroup$ – Kono Dio Da Sep 11 '17 at 20:30

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