What is the negation of the statement “$X$ is true if and only if $Y$ is true”? (There may be multiple ways to phrase this negation.)
The statement is equivalent to $X\iff Y$, which we have just seen is the negation of “either $X$ is true, or $Y$ is true, but not both”. And so the negation of the given statement is “either $X$ is true, or $Y$ is true, but not both”. Stated more simply, $X\ne Y$.
