ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nf3 GIF version

Theorem nf3 1604
Description: An alternate definition of df-nf 1395. (Contributed by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
nf3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Proof of Theorem nf3
StepHypRef Expression
1 nf2 1603 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 1430 . . . 4 𝑥𝑥𝜑
32nfri 1457 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
4319.21h 1494 . 2 (∀𝑥(∃𝑥𝜑𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
51, 4bitr4i 185 1 (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wnf 1394  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  eusv2nf  4278
  Copyright terms: Public domain W3C validator