Theorem re1luk2 1713

Description: luk-2 1658 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1luk2	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Proof of Theorem re1luk2

Step	Hyp	Ref	Expression
1		tbw-negdf 1701	. . . 4 ⊢ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
2		tbw-ax2 1703	. . . . 5 ⊢ ((((𝜑 → ⊥) → ¬ 𝜑) → ⊥) → ((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)))
3		tbwlem4 1710	. . . . 5 ⊢ (((((𝜑 → ⊥) → ¬ 𝜑) → ⊥) → ((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥))) → ((((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑)))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑))
5	1, 4	ax-mp 5	. . 3 ⊢ ((𝜑 → ⊥) → ¬ 𝜑)
6		tbw-ax1 1702	. . 3 ⊢ (((𝜑 → ⊥) → ¬ 𝜑) → ((¬ 𝜑 → 𝜑) → ((𝜑 → ⊥) → 𝜑)))
7	5, 6	ax-mp 5	. 2 ⊢ ((¬ 𝜑 → 𝜑) → ((𝜑 → ⊥) → 𝜑))
8		tbw-ax3 1704	. 2 ⊢ (((𝜑 → ⊥) → 𝜑) → 𝜑)
9	7, 8	tbwsyl 1706	1 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1550

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 210  df-tru 1541  df-fal 1551

This theorem is referenced by: (None)