Theorem tbw-negdf 1703

Description: The definition of negation, in terms of → and ⊥. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbw-negdf	⊢ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)

Proof of Theorem tbw-negdf

Step	Hyp	Ref	Expression
1		pm2.21 123	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
2		ax-1 6	. . . . 5 ⊢ (¬ 𝜑 → ((𝜑 → ⊥) → ¬ 𝜑))
3		falim 1556	. . . . 5 ⊢ (⊥ → ((𝜑 → ⊥) → ¬ 𝜑))
4	2, 3	ja 186	. . . 4 ⊢ ((𝜑 → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑))
5	4	pm2.43i 52	. . 3 ⊢ ((𝜑 → ⊥) → ¬ 𝜑)
6	1, 5	impbii 208	. 2 ⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))
7		tbw-bijust 1702	. 2 ⊢ ((¬ 𝜑 ↔ (𝜑 → ⊥)) ↔ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥))
8	6, 7	mpbi 229	1 ⊢ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ⊥wfal 1551

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

This theorem depends on definitions:  df-bi 206  df-tru 1542  df-fal 1552

This theorem is referenced by:  re1luk2  1715  re1luk3  1716