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

Step	Hyp	Ref	Expression
1		nfnd.1	. 2 ⊢ (φ → Ⅎxψ)
2		nfnf1 1790	. . 3 ⊢ ℲxℲxψ
3		df-nf 1545	. . . 4 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
4		hbnt 1775	. . . 4 ⊢ (∀x(ψ → ∀xψ) → (¬ ψ → ∀x ¬ ψ))
5	3, 4	sylbi 187	. . 3 ⊢ (Ⅎxψ → (¬ ψ → ∀x ¬ ψ))
6	2, 5	nfd 1766	. 2 ⊢ (Ⅎxψ → Ⅎx ¬ ψ)
7	1, 6	syl 15	1 ⊢ (φ → Ⅎx ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnf 1544

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-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  2613  nfneld  2614  nfrexd  2667