MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nf5-1 Structured version   Visualization version   GIF version

Theorem nf5-1 2188
Description: One direction of nf5 2290 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
Assertion
Ref Expression
nf5-1 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem nf5-1
StepHypRef Expression
1 exim 1918 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥𝑥𝜑))
2 hbe1a 2187 . . 3 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
31, 2syl6 35 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
43nfd 1870 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635  wex 1859  wnf 1863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-10 2184
This theorem depends on definitions:  df-bi 198  df-ex 1860  df-nf 1864
This theorem is referenced by:  nf5i  2189  nf5dh  2190  nf5dvOLD  2192  nf5d  2292  hbnt  2320
  Copyright terms: Public domain W3C validator