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Theorem bj-modald 32636
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
StepHypRef Expression
1 19.2 1890 . . 3 (∀𝑥𝜑 → ∃𝑥𝜑)
2 df-ex 1703 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
31, 2sylib 208 . 2 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
43con2i 134 1 (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-6 1886
This theorem depends on definitions:  df-bi 197  df-ex 1703
This theorem is referenced by: (None)
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