Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-nexdt Structured version   Visualization version   GIF version

Theorem bj-nexdt 36082
Description: Closed form of nexd 2206. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-nexdt (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))

Proof of Theorem bj-nexdt
StepHypRef Expression
1 nf5r 2179 . 2 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2 bj-nexdh 36012 . 2 (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → ¬ ∃𝑥𝜓)))
31, 2syl5com 31 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by:  bj-nexdvt  36083
  Copyright terms: Public domain W3C validator