Theorem bj-nexdh 33961

Description: Closed form of nexdh 1866 (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 1822	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ∀𝑥 ¬ 𝜓)))
2		alnex 1782	. 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
3	1, 2	syl8ib 258	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  bj-nexdh2  33962  bj-nexdt  34031