Theorem nfnt 1767

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.) df-nf 1701 changed. (Revised by Wolf Lammen, 4-Oct-2021.)

Assertion

Ref	Expression
nfnt	⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnt

Step	Hyp	Ref	Expression
1		notnot 135	. . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	alimi 1730	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
3	2	orim1i 538	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ ¬ 𝜑 ∨ ∀𝑥 ¬ 𝜑))
4		pm1.4 400	. . 3 ⊢ ((∀𝑥 ¬ ¬ 𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
5	3, 4	syl 17	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
6		nf3 1703	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
7		nf3 1703	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
8	5, 6, 7	3imtr4i 280	1 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  nfn  1768  nfnd  1769  nfan1  2056  19.23t  2066  wl-nfnbi  32493  wl-nfeqfb  32502  19.9alt  33270