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

Theorem bj-nnfe 34551
Description: Nonfreeness implies the equivalent of ax5e 1919. (Contributed by BJ, 28-Jul-2023.)
Assertion
Ref Expression
bj-nnfe (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))

Proof of Theorem bj-nnfe
StepHypRef Expression
1 df-bj-nnf 34544 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simplbi 501 1 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786  Ⅎ'wnnf 34543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400  df-bj-nnf 34544
This theorem is referenced by:  bj-nnfed  34552  bj-nnfei  34553  bj-nnfea  34554  bj-nnfim1  34564  bj-nnfim2  34565  bj-nnf-exlim  34576  bj-19.21t  34589  bj-19.36im  34591  bj-19.42t  34593  bj-sbft  34595
  Copyright terms: Public domain W3C validator