Theorem bj-modalbe

Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b . (Contributed by BJ, 20-Oct-2019)

	Ref	Expression
Assertion	bj-modalbe	$$⊢\phi \to \forall x\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		modal-b	$$⊢\phi \to \forall x\neg \forall x\neg \phi$$
2		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
3	2	biimpri	$$⊢\neg \forall x\neg \phi \to \exists x\phi$$
4	1 3	sylg	$$⊢\phi \to \forall x\exists x\phi$$