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

Theorem nf3 1662
Description: An alternate definition of df-nf 1454. (Contributed by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
nf3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Proof of Theorem nf3
StepHypRef Expression
1 nf2 1661 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 1489 . . . 4 𝑥𝑥𝜑
32nfri 1512 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
4319.21h 1550 . 2 (∀𝑥(∃𝑥𝜑𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
51, 4bitr4i 186 1 (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346  wnf 1453  wex 1485
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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454
This theorem is referenced by:  hbe1a  2016  eusv2nf  4441
  Copyright terms: Public domain W3C validator