Theorem nfnt 1679

Description: If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)

Assertion

Ref	Expression
nfnt	⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnt

Step	Hyp	Ref	Expression
1		nfnf1 1567	. 2 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2		df-nf 1484	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3		hbnt 1676	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4	2, 3	sylbi 121	. 2 ⊢ (Ⅎ𝑥𝜑 → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
5	1, 4	nfd 1546	1 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1371  Ⅎwnf 1483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1470  ax-gen 1472  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484

This theorem is referenced by:  nfnd  1680  nfn  1681