Theorem bj-nexdh 34809

Description: Closed form of nexdh 1868 (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

Step	Hyp	Ref	Expression
1		sylgt 1824	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ∀𝑥 ¬ 𝜓)))
2		alnex 1784	. 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
3	1, 2	syl8ib 255	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  bj-nexdh2  34810  bj-nexdt  34879