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Theorem bj-nnfa 36238
Description: Nonfreeness implies the equivalent of ax-5 1905. See nf5r 2182. (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 36234 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simprbi 495 1 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1773  Ⅎ'wnnf 36233
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 395  df-bj-nnf 36234
This theorem is referenced by:  bj-nnfad  36239  bj-nnfai  36240  bj-nnfea  36244  bj-nnfim1  36254  bj-nnfim2  36255  bj-nnf-alrim  36265  bj-19.23t  36280  bj-19.37im  36282  bj-19.42t  36283  bj-sbft  36285
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