This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem axc7e

Description: Abbreviated version of axc7 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 8-Jul-2022)

		Ref	Expression
	Assertion	axc7e	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
2	1	19.21bi	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → 𝜑 )