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Theorem modal-b 2140
Description: The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
Assertion
Ref Expression
modal-b (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Proof of Theorem modal-b
StepHypRef Expression
1 axc7 2130 . 2 (¬ ∀𝑥 ¬ ∀𝑥 ¬ 𝜑 → ¬ 𝜑)
21con4i 113 1 (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1479
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-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-ex 1703
This theorem is referenced by:  bj-modalbe  32653
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