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

Theorem bj-hbntbi 33548
Description: Strengthening hbnt 2228 by replacing its succedent with a biconditional. See also hbntg 32571 and hbntal 40306. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 33547. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbntbi (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))

Proof of Theorem bj-hbntbi
StepHypRef Expression
1 bj-19.9htbi 33547 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
21bicomd 215 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑))
32notbid 310 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
4 alnex 1744 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
53, 4syl6bbr 281 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1505  wex 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-ex 1743
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator