Theorem tbwlem1 1705

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
tbwlem1	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem tbwlem1

Step	Hyp	Ref	Expression
1		tbw-ax1 1700	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)))
2		tbw-ax2 1701	. . . . 5 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜓))
3		tbw-ax1 1700	. . . . 5 ⊢ (((𝜓 → 𝜒) → 𝜓) → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
4	2, 3	tbwsyl 1704	. . . 4 ⊢ (𝜓 → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
5		tbw-ax1 1700	. . . . 5 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
6		tbw-ax3 1702	. . . . 5 ⊢ (((((𝜓 → 𝜒) → 𝜒) → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒))
7	5, 6	tbwsyl 1704	. . . 4 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒))
8	4, 7	tbwsyl 1704	. . 3 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜒))
9		tbw-ax1 1700	. . 3 ⊢ ((𝜓 → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))))
10	8, 9	ax-mp 5	. 2 ⊢ ((((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
11	1, 10	tbwsyl 1704	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  tbwlem2  1706  tbwlem4  1708  tbwlem5  1709  re1luk3  1712