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Theorem bj-nnfnfTEMP 34547
Description: New nonfreeness implies old nonfreeness on minimal implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1791 except via df-nf 1791 directly. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nnfnfTEMP (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem bj-nnfnfTEMP
StepHypRef Expression
1 bj-nnfea 34543 . 2 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2 df-nf 1791 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
31, 2sylibr 237 1 (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786  wnf 1790  Ⅎ'wnnf 34532
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-nf 1791  df-bj-nnf 34533
This theorem is referenced by: (None)
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