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Theorem bj-modalbe 31865
Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2127. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-modalbe (𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem bj-modalbe
StepHypRef Expression
1 modal-b 2127 . 2 (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
2 df-ex 1696 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
32biimpri 217 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑)
41, 3sylg 1740 1 (𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1473  wex 1695
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: (None)
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