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Theorem nfcd 2215
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 1456 . 2 (𝜑 → ∀𝑦𝑥 𝑦𝐴)
4 df-nfc 2209 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4sylibr 132 1 (𝜑𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390  wcel 1434  wnfc 2207
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  df-nfc 2209
This theorem is referenced by:  nfabd  2238  dvelimdc  2239  sbnfc2  2963
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