Theorem tbwlem2 1709

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
tbwlem2	⊢ ((𝜑 → (𝜓 → ⊥)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))

Proof of Theorem tbwlem2

Step	Hyp	Ref	Expression
1		tbw-ax4 1706	. . . . 5 ⊢ (⊥ → 𝜒)
2		tbw-ax1 1703	. . . . . 6 ⊢ ((𝜓 → ⊥) → ((⊥ → 𝜒) → (𝜓 → 𝜒)))
3		tbwlem1 1708	. . . . . 6 ⊢ (((𝜓 → ⊥) → ((⊥ → 𝜒) → (𝜓 → 𝜒))) → ((⊥ → 𝜒) → ((𝜓 → ⊥) → (𝜓 → 𝜒))))
4	2, 3	ax-mp 5	. . . . 5 ⊢ ((⊥ → 𝜒) → ((𝜓 → ⊥) → (𝜓 → 𝜒)))
5	1, 4	ax-mp 5	. . . 4 ⊢ ((𝜓 → ⊥) → (𝜓 → 𝜒))
6		tbwlem1 1708	. . . 4 ⊢ (((𝜓 → ⊥) → (𝜓 → 𝜒)) → (𝜓 → ((𝜓 → ⊥) → 𝜒)))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝜓 → ((𝜓 → ⊥) → 𝜒))
8		tbw-ax1 1703	. . 3 ⊢ ((𝜑 → (𝜓 → ⊥)) → (((𝜓 → ⊥) → 𝜒) → (𝜑 → 𝜒)))
9		tbw-ax1 1703	. . 3 ⊢ ((𝜓 → ((𝜓 → ⊥) → 𝜒)) → ((((𝜓 → ⊥) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))))
10	7, 8, 9	mpsyl 68	. 2 ⊢ ((𝜑 → (𝜓 → ⊥)) → (𝜓 → (𝜑 → 𝜒)))
11		tbw-ax1 1703	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))
12	10, 11	tbwsyl 1707	1 ⊢ ((𝜑 → (𝜓 → ⊥)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))

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:  tbwlem4  1711