Theorem tbwlem5 1712

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbwlem5	⊢ (((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑)

Proof of Theorem tbwlem5

Step	Hyp	Ref	Expression
1		tbw-ax2 1704	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜑))
2		tbw-ax1 1703	. . . 4 ⊢ ((𝜓 → 𝜑) → ((𝜑 → ⊥) → (𝜓 → ⊥)))
3	1, 2	tbwsyl 1707	. . 3 ⊢ (𝜑 → ((𝜑 → ⊥) → (𝜓 → ⊥)))
4		tbwlem1 1708	. . 3 ⊢ ((𝜑 → ((𝜑 → ⊥) → (𝜓 → ⊥))) → ((𝜑 → ⊥) → (𝜑 → (𝜓 → ⊥))))
5	3, 4	ax-mp 5	. 2 ⊢ ((𝜑 → ⊥) → (𝜑 → (𝜓 → ⊥)))
6		tbwlem4 1711	. 2 ⊢ (((𝜑 → ⊥) → (𝜑 → (𝜓 → ⊥))) → (((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑))
7	5, 6	ax-mp 5	1 ⊢ (((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑)

Syntax hints:   → wi 4  ⊥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:  re1luk3  1715