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

Theorem bj-nnfa 35125
Description: Nonfreeness implies the equivalent of ax-5 1913. See nf5r 2187. (Contributed by BJ, 28-Jul-2023.)
Assertion
Ref Expression
bj-nnfa (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-nnfa
StepHypRef Expression
1 df-bj-nnf 35121 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simprbi 497 1 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539  wex 1781  Ⅎ'wnnf 35120
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 206  df-an 397  df-bj-nnf 35121
This theorem is referenced by:  bj-nnfad  35126  bj-nnfai  35127  bj-nnfea  35131  bj-nnfim1  35141  bj-nnfim2  35142  bj-nnf-alrim  35152  bj-19.23t  35167  bj-19.37im  35169  bj-19.42t  35170  bj-sbft  35172
  Copyright terms: Public domain W3C validator