Theorem hbe1

Description: The setvar x is not free in E. x ph . Corresponds to the axiom (5) of modal logic (see also modal5 ). (Contributed by NM, 24-Jan-1993)

	Ref	Expression
Assertion	hbe1	$$⊢\exists x\phi \to \forall x\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
2		hbn1	$$⊢\neg \forall x\neg \phi \to \forall x\neg \forall x\neg \phi$$
3	1 2	hbxfrbi	$$⊢\exists x\phi \to \forall x\exists x\phi$$