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Theorem nfcd 2746
 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 2069 . 2 (𝜑 → ∀𝑦𝑥 𝑦𝐴)
4 df-nfc 2740 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4sylibr 223 1 (𝜑𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701  df-nfc 2740 This theorem is referenced by:  nfabd2  2770  dvelimdc  2772  sbnfc2  3959
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