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Theorem bj-nnfe 36116
Description: Nonfreeness implies the equivalent of ax5e 1907. (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 36109 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simplbi 497 1 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1773  Ⅎ'wnnf 36108
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 396  df-bj-nnf 36109
This theorem is referenced by:  bj-nnfed  36117  bj-nnfei  36118  bj-nnfea  36119  bj-nnfim1  36129  bj-nnfim2  36130  bj-nnf-exlim  36141  bj-19.21t  36154  bj-19.36im  36156  bj-19.42t  36158  bj-sbft  36160
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