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Theorem bj-nnfa 34136
Description: Nonfreeness implies the equivalent of ax-5 1911. See nf5r 2194, nf5ri 2196. (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 34132 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
21simprbi 500 1 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  Ⅎ'wnnf 34131
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 210  df-an 400  df-bj-nnf 34132
This theorem is referenced by:  bj-nnfad  34137  bj-nnfea  34140  bj-nnfim1  34149  bj-nnfim2  34150  bj-nnf-alrim  34160  bj-19.23t  34175  bj-19.37im  34177  bj-19.42t  34178  bj-sbft  34180
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