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Theorem nfcr 2215
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 2212 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
2 sp 1442 . 2  |-  ( A. y F/ x  y  e.  A  ->  F/ x  y  e.  A )
31, 2sylbi 119 1  |-  ( F/_ x A  ->  F/ x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   F/wnf 1390    e. wcel 1434   F/_wnfc 2210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nfc 2212
This theorem is referenced by:  nfcrii  2216  nfcrd  2236  abidnf  2771  csbtt  2929  csbnestgf  2965
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