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Metamath Proof Explorer

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	$⊢ \exists x φ \to \forall x \exists x φ$

Proof

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