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Theorem bj-nexdh 31790
Description: Closed form of nexdh 1779 (and more general since it uses 𝜒). (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 1739 . 2 (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ∀𝑥 ¬ 𝜓)))
2 alnex 1697 . 2 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
31, 2syl8ib 245 1 (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1728
This theorem depends on definitions:  df-bi 196  df-ex 1696
This theorem is referenced by:  bj-nexdh2  31791  bj-nexdt  31874
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