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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	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
2		hbn1	⊢ ( ¬ ∀ 𝑥 ¬ 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 ¬ 𝜑 )
3	1 2	hbxfrbi	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )