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Theorem nfcr 2905
 Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
nfcr (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcr
StepHypRef Expression
1 df-nfc 2902 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 sp 2207 . 2 (∀𝑦𝑥 𝑦𝐴 → Ⅎ𝑥 𝑦𝐴)
31, 2sylbi 207 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1629  Ⅎwnf 1856   ∈ wcel 2145  Ⅎwnfc 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203 This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nfc 2902 This theorem is referenced by:  nfcrii  2906  nfcrd  2920  nfnfc  2923  abidnf  3527  csbtt  3693  csbnestgf  4140
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