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Theorem nfcr 2964
 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 2961 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 sp 2175 . 2 (∀𝑦𝑥 𝑦𝐴 → Ⅎ𝑥 𝑦𝐴)
31, 2sylbi 219 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1529  Ⅎwnf 1778   ∈ wcel 2108  Ⅎwnfc 2959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-12 2170 This theorem depends on definitions:  df-bi 209  df-ex 1775  df-nfc 2961 This theorem is referenced by:  nfcriv  2965  nfcrd  2967  abidnf  3692  csbtt  3898  csbnestgfw  4369  csbnestgf  4374  wl-dfralf  34831
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