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 34167
 Description: Strengthening hbnt 2298 by replacing its succedent with a biconditional. See also hbntg 33175 and hbntal 41274. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 34166. (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 34166 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
21bicomd 226 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑))
32notbid 321 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
4 alnex 1783 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
53, 4bitr4di 292 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator