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Theorem nfcd 2484
 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 1765 . 2 (φyx y A)
4 df-nfc 2478 . 2 (xAyx y A)
53, 4sylibr 203 1 (φxA)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-nfc 2478 This theorem is referenced by:  nfabd2  2507  dvelimdc  2509  sbnfc2  3196
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