Theorem bj-modald 34756

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 1985	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2		df-ex 1788	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	sylib 221	. 2 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
4	3	con2i 141	1 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-6 1976

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by: (None)