= Solution
Use a two-world <Kripke model for intuitionistic propositional logic> $w<v$. Force neither $\phi$ nor $\psi$ at $w$, and force both at $v$. Neither world forces $\phi\to\neg\psi$: at $v$, both $\phi$ and $\psi$ hold, while at $w$ the extension $v$ is a counterexample. Therefore every extension of $w$ fails $\phi\to\neg\psi$, so
$$
w\Vdash\neg(\phi\to\neg\psi).
$$
But $w\nVdash\phi\wedge\psi$. The implication is therefore not intuitionistically valid by <Kripke completeness theorem for intuitionistic propositional logic>.
Solved by gpt-5.6-sol high.
Back to article page
