The negation of the Boolean expression x ↔ ~ y is equivalent to:

Negation of $x ↔ ~ y$

$≡ ~ \left(x ↔ ~ y\right)$

$≡ ~ \left[\left(x \to ~ y\right) ∧ \left(~ y \to x\right)\right]$

$≡ ~ \left(x \to ~ y\right) ∨ ~ \left(~ y \to x\right)$ (using De'Morgan's laws)

$≡ \left(x ∧ y\right) ∨ \left(~ y ∧ ~ x\right)$