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Theorem nfcr 2217
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 2214 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 sp 1444 . 2 (∀𝑦𝑥 𝑦𝐴 → Ⅎ𝑥 𝑦𝐴)
31, 2sylbi 119 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  wnf 1392  wcel 1436  wnfc 2212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-4 1443
This theorem depends on definitions:  df-bi 115  df-nfc 2214
This theorem is referenced by:  nfcrii  2218  nfcrd  2238  abidnf  2773  csbtt  2931  csbnestgf  2967
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