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Theorem bj-nnfa 36729
Description: Nonfreeness implies the equivalent of ax-5 1910. See nf5r 2194. (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 36725 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simprbi 496 1 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wex 1779  Ⅎ'wnnf 36724
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 207  df-an 396  df-bj-nnf 36725
This theorem is referenced by:  bj-nnfad  36730  bj-nnfai  36731  bj-nnfea  36735  bj-nnfim1  36745  bj-nnfim2  36746  bj-nnf-alrim  36756  bj-19.23t  36771  bj-19.37im  36773  bj-19.42t  36774  bj-sbft  36776
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