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

Theorem nfcr 2582
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
nfcr  |-  ( F/_ x A  ->  F/ x  y  e.  A )
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem nfcr
StepHypRef Expression
1 df-nfc 2579 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
2 sp 1912 . 2  |-  ( A. y F/ x  y  e.  A  ->  F/ x  y  e.  A )
31, 2sylbi 198 1  |-  ( F/_ x A  ->  F/ x  y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   F/wnf 1663    e. wcel 1870   F/_wnfc 2577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nfc 2579
This theorem is referenced by:  nfcrii  2583  nfcrd  2597  nfnfc  2600  abidnf  3246  csbtt  3412  csbnestgf  3819  bj-nfcrii  31211
  Copyright terms: Public domain W3C validator