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Theorem bj-modalbe 33919
Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2329. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-modalbe (𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem bj-modalbe
StepHypRef Expression
1 modal-b 2329 . 2 (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
2 df-ex 1772 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
32biimpri 229 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑)
41, 3sylg 1814 1 (𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772
This theorem is referenced by:  bj-modal4  33945  bj-19.12  33987
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