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Theorem nfcr 2411
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 2408 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
2 sp 1716 . 2  |-  ( A. y F/ x  y  e.  A  ->  F/ x  y  e.  A )
31, 2sylbi 187 1  |-  ( F/_ x A  ->  F/ x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   F/wnf 1531    e. wcel 1684   F/_wnfc 2406
This theorem is referenced by:  nfcrii  2412  nfcrd  2432  abidnf  2934  csbtt  3093  csbnestgf  3129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-nfc 2408
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