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Theorem nfcr 2753
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 2750 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 sp 2051 . 2 (∀𝑦𝑥 𝑦𝐴 → Ⅎ𝑥 𝑦𝐴)
31, 2sylbi 207 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
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-nfc 2750
This theorem is referenced by:  nfcrii  2754  nfcrd  2767  nfnfc  2770  abidnf  3357  csbtt  3525  csbnestgf  3968  bj-nfcrii  32495
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