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Theorem nfcr 2566
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 2563 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
2 sp 1764 . 2  |-  ( A. y F/ x  y  e.  A  ->  F/ x  y  e.  A )
31, 2sylbi 189 1  |-  ( F/_ x A  ->  F/ x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   F/wnf 1554    e. wcel 1726   F/_wnfc 2561
This theorem is referenced by:  nfcrii  2567  nfcrd  2587  abidnf  3105  csbtt  3265  csbnestgf  3301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-ex 1552  df-nfc 2563
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