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Theorem bj-hbntbi 36090
Description: Strengthening hbnt 2282 by replacing its consequent with a biconditional. See also hbntg 35310 and hbntal 43890. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 36089. (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 36089 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
21bicomd 222 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑))
32notbid 318 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
4 alnex 1775 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
53, 4bitr4di 289 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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