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Theorem nf5di 2117
 Description: Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)
Hypothesis
Ref Expression
nf5di.1 (𝜑 → Ⅎ𝑥𝜑)
Assertion
Ref Expression
nf5di 𝑥𝜑

Proof of Theorem nf5di
StepHypRef Expression
1 nf5di.1 . . . 4 (𝜑 → Ⅎ𝑥𝜑)
21nf5rd 2064 . . 3 (𝜑 → (𝜑 → ∀𝑥𝜑))
32pm2.43i 52 . 2 (𝜑 → ∀𝑥𝜑)
43nf5i 2022 1 𝑥𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479  Ⅎwnf 1706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045 This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708 This theorem is referenced by: (None)
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