TITLE: Is the negation of a non-theorem a theorem?
QUESTION [19 upvotes]: I don't know if that is something obvious or if it is a dumb question. But it seems to be true.
Consider the non-theorem $\forall x. x < 1$. Its negation is $\exists x. x \geq 1$ and is a theorem.
Is this always true?
I couldn't find a counterexample.
If this is true I would like to know a good explanation why.

REPLY [6 votes]: Suppose we are talking about the theory of groups.  The statement
$$
\forall x \forall y \big(xy=yx\big)
$$
is not a theorem (it is false for some groups).  It negation
$$
\exists x \exists y \big(xy \ne yx\big)
$$
is also not a theorem (it, too, is false for some groups).