MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfcr Structured version   Visualization version   GIF version

Theorem nfcr 2755
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 2752 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 sp 2052 . 2 (∀𝑦𝑥 𝑦𝐴 → Ⅎ𝑥 𝑦𝐴)
31, 2sylbi 207 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1480  wnf 1707  wcel 1989  wnfc 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-ex 1704  df-nfc 2752
This theorem is referenced by:  nfcrii  2756  nfcrd  2770  nfnfc  2773  abidnf  3373  csbtt  3542  csbnestgf  3994  bj-nfcrii  32835
  Copyright terms: Public domain W3C validator