Theorem re1luk3 1712

Description: luk-3 1657 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1710 and re1luk2 1711 proves that tbw-ax1 1700, tbw-ax2 1701, tbw-ax3 1702, and tbw-ax4 1703, with ax-mp 5 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1luk3	⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Proof of Theorem re1luk3

Step	Hyp	Ref	Expression
1		tbw-ax4 1703	. . . 4 ⊢ (⊥ → 𝜓)
2		tbw-ax1 1700	. . . . 5 ⊢ ((𝜑 → ⊥) → ((⊥ → 𝜓) → (𝜑 → 𝜓)))
3		tbwlem1 1705	. . . . 5 ⊢ (((𝜑 → ⊥) → ((⊥ → 𝜓) → (𝜑 → 𝜓))) → ((⊥ → 𝜓) → ((𝜑 → ⊥) → (𝜑 → 𝜓))))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((⊥ → 𝜓) → ((𝜑 → ⊥) → (𝜑 → 𝜓)))
5	1, 4	ax-mp 5	. . 3 ⊢ ((𝜑 → ⊥) → (𝜑 → 𝜓))
6		tbwlem1 1705	. . 3 ⊢ (((𝜑 → ⊥) → (𝜑 → 𝜓)) → (𝜑 → ((𝜑 → ⊥) → 𝜓)))
7	5, 6	ax-mp 5	. 2 ⊢ (𝜑 → ((𝜑 → ⊥) → 𝜓))
8		tbw-negdf 1699	. . . 4 ⊢ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
9		tbwlem5 1709	. . . 4 ⊢ ((((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) → (¬ 𝜑 → (𝜑 → ⊥)))
10	8, 9	ax-mp 5	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
11		tbw-ax1 1700	. . 3 ⊢ ((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → 𝜓) → (¬ 𝜑 → 𝜓)))
12	10, 11	ax-mp 5	. 2 ⊢ (((𝜑 → ⊥) → 𝜓) → (¬ 𝜑 → 𝜓))
13	7, 12	tbwsyl 1704	1 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1552

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 207  df-tru 1543  df-fal 1553

This theorem is referenced by: (None)