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Theorem bj-dfnnf3 34707
Description: Alternate definition of nonfreeness when sp 2182 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1792. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dfnnf3 (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-dfnnf3
StepHypRef Expression
1 bj-nnfea 34684 . 2 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2 bj-19.21bit 34640 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
3 bj-19.23bit 34641 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
4 df-bj-nnf 34674 . . 3 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
52, 3, 4sylanbrc 586 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
61, 5impbii 212 1 (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541  wex 1787  Ⅎ'wnnf 34673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2177
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-bj-nnf 34674
This theorem is referenced by:  bj-nfnnfTEMP  34708
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