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Theorem bj-nexdt 32985
Description: Closed form of nexd 2228. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-nexdt (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))

Proof of Theorem bj-nexdt
StepHypRef Expression
1 nf5r 2203 . 2 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2 bj-nexdh 32904 . 2 (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → ¬ ∃𝑥𝜓)))
31, 2syl5com 31 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1622  wex 1845  wnf 1849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-12 2188
This theorem depends on definitions:  df-bi 197  df-ex 1846  df-nf 1851
This theorem is referenced by:  bj-nexdvt  32986
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