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Theorem bj-nexdh 32245
Description: Closed form of nexdh 1789 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-nexdh (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Proof of Theorem bj-nexdh
StepHypRef Expression
1 sylgt 1746 . 2 (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ∀𝑥 ¬ 𝜓)))
2 alnex 1703 . 2 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
31, 2syl8ib 246 1 (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by:  bj-nexdh2  32246  bj-nexdt  32326
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