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

Theorem nfnt 1587
Description: If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)
Assertion
Ref Expression
nfnt (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnt
StepHypRef Expression
1 nfnf1 1477 . 2 𝑥𝑥𝜑
2 df-nf 1391 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3 hbnt 1584 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
42, 3sylbi 119 . 2 (Ⅎ𝑥𝜑 → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
51, 4nfd 1457 1 (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1283  wnf 1390
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-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391
This theorem is referenced by:  nfnd  1588  nfn  1589
  Copyright terms: Public domain W3C validator