Theorem bj-modald

Description: A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021)

	Ref	Expression
Assertion	bj-modald	$$⊢\forall x\neg \phi \to \neg \forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		19.2	$$⊢\forall x\phi \to \exists x\phi$$
2		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
3	1 2	sylib	$$⊢\forall x\phi \to \neg \forall x\neg \phi$$
4	3	con2i	$$⊢\forall x\neg \phi \to \neg \forall x\phi$$