Theorem nfntOLDOLD 1783

Description: Obsolete proof of nfnt 1782 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1710 changed. (Revised by Wolf Lammen, 4-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nfntOLDOLD	⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfntOLDOLD

Step	Hyp	Ref	Expression
1		notnot 136	. . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	alimi 1739	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
3	2	orim1i 539	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ ¬ 𝜑 ∨ ∀𝑥 ¬ 𝜑))
4		pm1.4 401	. . 3 ⊢ ((∀𝑥 ¬ ¬ 𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
5	3, 4	syl 17	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
6		nf3 1712	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
7		nf3 1712	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
8	5, 6, 7	3imtr4i 281	1 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)