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Theorem nfcd 2598
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 1966 . 2  |-  ( ph  ->  A. y F/ x  y  e.  A )
4 df-nfc 2592 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
53, 4sylibr 217 1  |-  ( ph  -> 
F/_ x A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1453   F/wnf 1678    e. wcel 1898   F/_wnfc 2590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-12 1944
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-nfc 2592
This theorem is referenced by:  nfabd2  2622  dvelimdc  2624  sbnfc2  3808
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