Theorem bj-modalb

Description: A short form of the axiom B of modal logic using only primitive symbols ( -> , -. , A. ). (Contributed by BJ, 4-Apr-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-modalb	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		axc7	⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → 𝜑 )
2	1	con1i	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )