Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-modalbe Structured version   Visualization version   GIF version

Theorem bj-modalbe 32803
Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2180. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-modalbe (𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem bj-modalbe
StepHypRef Expression
1 modal-b 2180 . 2 (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
2 df-ex 1745 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
32biimpri 218 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑)
41, 3sylg 1790 1 (𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator