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Theorem nfnd 1791
 Description: If in a context x is not free in ψ, it is not free in ¬ ψ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
Hypothesis
Ref Expression
nfnd.1 (φ → Ⅎxψ)
Assertion
Ref Expression
nfnd (φ → Ⅎx ¬ ψ)

Proof of Theorem nfnd
StepHypRef Expression
1 nfnd.1 . 2 (φ → Ⅎxψ)
2 nfnf1 1790 . . 3 xxψ
3 df-nf 1545 . . . 4 (Ⅎxψx(ψxψ))
4 hbnt 1775 . . . 4 (x(ψxψ) → (¬ ψx ¬ ψ))
53, 4sylbi 187 . . 3 (Ⅎxψ → (¬ ψx ¬ ψ))
62, 5nfd 1766 . 2 (Ⅎxψ → Ⅎx ¬ ψ)
71, 6syl 15 1 (φ → Ⅎx ¬ ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfn  1793  nfand  1822  nfbidOLD  1833  nfexd  1854  19.9tOLD  1857  nfexd2  1973  cbvexd  2009  nfned  2612  nfneld  2613  nfrexd  2666
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