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Theorem bj-modalbe 33031
Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2310. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-modalbe (𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem bj-modalbe
StepHypRef Expression
1 modal-b 2310 . 2 (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
2 df-ex 1856 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
32biimpri 219 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑)
41, 3sylg 1901 1 (𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1632  wex 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-10 2177  ax-12 2206
This theorem depends on definitions:  df-bi 198  df-ex 1856
This theorem is referenced by: (None)
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