Theorem bj-modald 37353

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

Assertion

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

Proof of Theorem bj-modald

Step	Hyp	Ref	Expression
1		19.2 2009	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2		df-ex 1813	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	sylib 221	. 2 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
4	3	con2i 140	1 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)