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

Theorem nfcd 2277
 Description: Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfcd.1 𝑦𝜑
nfcd.2 (𝜑 → Ⅎ𝑥 𝑦𝐴)
Assertion
Ref Expression
nfcd (𝜑𝑥𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfcd
StepHypRef Expression
1 nfcd.1 . . 3 𝑦𝜑
2 nfcd.2 . . 3 (𝜑 → Ⅎ𝑥 𝑦𝐴)
31, 2alrimi 1503 . 2 (𝜑 → ∀𝑦𝑥 𝑦𝐴)
4 df-nfc 2271 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4sylibr 133 1 (𝜑𝑥𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330  Ⅎwnf 1437   ∈ wcel 1481  Ⅎwnfc 2269 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 1424  ax-gen 1426  ax-4 1488 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-nfc 2271 This theorem is referenced by:  nfabd  2301  dvelimdc  2302  sbnfc2  3065
 Copyright terms: Public domain W3C validator