ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nf2 GIF version

Theorem nf2 1647
Description: An alternate definition of df-nf 1438, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
nf2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem nf2
StepHypRef Expression
1 df-nf 1438 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nfa1 1522 . . . 4 𝑥𝑥𝜑
32nfri 1500 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
4319.23h 1475 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
51, 4bitri 183 1 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1330  wnf 1437  wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  nf3  1648  nf4dc  1649  nf4r  1650  eusv2i  4380
  Copyright terms: Public domain W3C validator