Theorem bj-modald

Description: A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021)

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

Proof

Step	Hyp	Ref	Expression
1		19.2	⊢ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜑 )
2		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
3	1 2	sylib	⊢ ( ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 ¬ 𝜑 )
4	3	con2i	⊢ ( ∀ 𝑥 ¬ 𝜑 → ¬ ∀ 𝑥 𝜑 )