Theorem nfnbi 1851

Description: A variable is non-free in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.)

Assertion

Ref	Expression
nfnbi	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnbi

Step	Hyp	Ref	Expression
1		orcom 866	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
2		nf3 1783	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
3		nf3 1783	. . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
4		notnotb 317	. . . . 5 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
5	4	albii 1816	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
6	5	orbi2i 909	. . 3 ⊢ ((∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
7	3, 6	bitr4i 280	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
8	1, 2, 7	3bitr4i 305	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∨ wo 843  ∀wal 1531  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781

This theorem is referenced by:  nfnt  1852  wl-sb8et  34783