Theorem bj-nexdvt 32688

Description: Closed form of nexdv 1864. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-nexdvt	⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem bj-nexdvt

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-nexdt 32687	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)