Theorem bj-nnsn 13614

Description: As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnsn	⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))

Proof of Theorem bj-nnsn

Step	Hyp	Ref	Expression
1		con3 632	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) → (¬ ¬ 𝜓 → ¬ 𝜑))
2	1	con3d 621	. . 3 ⊢ ((𝜑 → ¬ 𝜓) → (¬ ¬ 𝜑 → ¬ ¬ ¬ 𝜓))
3		notnotnot 624	. . 3 ⊢ (¬ ¬ ¬ 𝜓 ↔ ¬ 𝜓)
4	2, 3	syl6ib 160	. 2 ⊢ ((𝜑 → ¬ 𝜓) → (¬ ¬ 𝜑 → ¬ 𝜓))
5		notnot 619	. . 3 ⊢ (𝜑 → ¬ ¬ 𝜑)
6	5	imim1i 60	. 2 ⊢ ((¬ ¬ 𝜑 → ¬ 𝜓) → (𝜑 → ¬ 𝜓))
7	4, 6	impbii 125	1 ⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)