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Theorem bj-hbntbi 33026
 Description: Strengthening hbnt 2308 by replacing its succedent with a biconditional. See also hbntg 32041 and hbntal 39288. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 33025. (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 33025 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
21bicomd 213 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑))
32notbid 307 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
4 alnex 1853 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
53, 4syl6bbr 278 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1628  ∃wex 1851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202 This theorem depends on definitions:  df-bi 197  df-ex 1852 This theorem is referenced by: (None)
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