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Theorem bj-nnfa 35222
Description: Nonfreeness implies the equivalent of ax-5 1914. See nf5r 2188. (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 35218 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simprbi 498 1 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1782  Ⅎ'wnnf 35217
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 398  df-bj-nnf 35218
This theorem is referenced by:  bj-nnfad  35223  bj-nnfai  35224  bj-nnfea  35228  bj-nnfim1  35238  bj-nnfim2  35239  bj-nnf-alrim  35249  bj-19.23t  35264  bj-19.37im  35266  bj-19.42t  35267  bj-sbft  35269
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