Theorem nfnbiOLD 1859

Description: Obsolete version of nfnbi 1858 as of 6-Oct-2024. (Contributed by BJ, 6-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nfnbiOLD	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnbiOLD

Step	Hyp	Ref	Expression
1		orcom 866	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
2		nf3 1790	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
3		nf3 1790	. . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
4		notnotb 314	. . . . 5 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
5	4	albii 1823	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
6	5	orbi2i 909	. . 3 ⊢ ((∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
7	3, 6	bitr4i 277	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
8	1, 2, 7	3bitr4i 302	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∨ wo 843  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)