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Theorem nfcd 2756
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 2080 . 2 (𝜑 → ∀𝑦𝑥 𝑦𝐴)
4 df-nfc 2750 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4sylibr 224 1 (𝜑𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wnf 1705  wcel 1987  wnfc 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707  df-nfc 2750
This theorem is referenced by:  nfabd2  2780  dvelimdc  2782  sbnfc2  3985
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