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Theorem nfcd 2218
Description: Deduce that a class  A does not have  x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfcd.1  |-  F/ y
ph
nfcd.2  |-  ( ph  ->  F/ x  y  e.  A )
Assertion
Ref Expression
nfcd  |-  ( ph  -> 
F/_ x A )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem nfcd
StepHypRef Expression
1 nfcd.1 . . 3  |-  F/ y
ph
2 nfcd.2 . . 3  |-  ( ph  ->  F/ x  y  e.  A )
31, 2alrimi 1456 . 2  |-  ( ph  ->  A. y F/ x  y  e.  A )
4 df-nfc 2212 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
53, 4sylibr 132 1  |-  ( ph  -> 
F/_ x 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-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-nfc 2212
This theorem is referenced by:  nfabd  2241  dvelimdc  2242  sbnfc2  2971
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