MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nf5di Structured version   Visualization version   GIF version

Theorem nf5di 2282
Description: Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either 𝑥𝜑 (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 2220 . . 3 (𝜑 → (𝜑 → ∀𝑥𝜑))
32pm2.43i 52 . 2 (𝜑 → ∀𝑥𝜑)
43nf5i 2179 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator