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Theorem nfcd 2170
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 yφ
nfcd.2 (φ → Ⅎx y A)
Assertion
Ref Expression
nfcd (φxA)
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem nfcd
StepHypRef Expression
1 nfcd.1 . . 3 yφ
2 nfcd.2 . . 3 (φ → Ⅎx y A)
31, 2alrimi 1412 . 2 (φyx y A)
4 df-nfc 2164 . 2 (xAyx y A)
53, 4sylibr 137 1 (φxA)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  wnf 1346   wcel 1390  wnfc 2162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-nfc 2164
This theorem is referenced by:  nfabd  2193  dvelimdc  2194  sbnfc2  2900
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