Theorem bj-nexdt 32687

Description: Closed form of nexd 2089. (Contributed by BJ, 20-Oct-2019.)

Assertion

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

Proof of Theorem bj-nexdt

Step	Hyp	Ref	Expression
1		nf5r 2064	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		bj-nexdh 32606	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → ¬ ∃𝑥𝜓)))
3	1, 2	syl5com 31	1 ⊢ (Ⅎ𝑥𝜑 → (∀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:  bj-nexdvt  32688