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Theorem modal-b 2127
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 2117 . 2 (¬ ∀𝑥 ¬ ∀𝑥 ¬ 𝜑 → ¬ 𝜑)
21con4i 112 1 (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-ex 1696
This theorem is referenced by:  bj-modalbe  31865
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