Theorem bj-nexdt 34879

Description: Closed form of nexd 2214. (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 2187	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		bj-nexdh 34809	. 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 1537  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by:  bj-nexdvt  34880