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

Theorem nfd 1457
Description: Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
nfd.1 𝑥𝜑
nfd.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
Assertion
Ref Expression
nfd (𝜑 → Ⅎ𝑥𝜓)

Proof of Theorem nfd
StepHypRef Expression
1 nfd.1 . . . 4 𝑥𝜑
21nfri 1453 . . 3 (𝜑 → ∀𝑥𝜑)
3 nfd.2 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
42, 3alrimih 1399 . 2 (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
5 df-nf 1391 . 2 (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
64, 5sylibr 132 1 (𝜑 → Ⅎ𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  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-5 1377  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  nfdh  1458  nfrimi  1459  nfnt  1587  cbv1h  1674  nfald  1684  a16nf  1788  dvelimALT  1928  dvelimfv  1929  nfsb4t  1932  hbeud  1964
  Copyright terms: Public domain W3C validator