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Theorem nfcd 2895
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 2206 . 2 (𝜑 → ∀𝑦𝑥 𝑦𝐴)
4 df-nfc 2889 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4sylibr 233 1 (𝜑𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wnf 1786  wcel 2106  wnfc 2887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787  df-nfc 2889
This theorem is referenced by:  nfabdw  2930  nfabdwOLD  2931  nfabd  2932  dvelimdc  2934  nfcvf  2936  sbnfc2  4370
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