Theorem nic-luk1

Description: Proof of luk-1 from nic-ax and nic-mp (and Definitions nic-dfim and nic-dfneg ). Note that the standard axioms ax-1 , ax-2 , and ax-3 are proved from the Lukasiewicz axioms by Theorems ax1 , ax2 , and ax3 . (Contributed by Jeff Hoffman, 18-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nic-luk1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nic-dfim	⊢ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 → 𝜓 ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) ⊼ ( 𝜑 → 𝜓 ) ) ) )
2	1	nic-bi2	⊢ ( ( 𝜑 → 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ) )
3		nic-ax	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ⊼ ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) ) )
4	3	nic-isw2	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) ⊼ ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ) )
5	4	nic-idel	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) ⊼ ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) ) )
6		nic-dfim	⊢ ( ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 → 𝜒 ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) )
7	6	nic-bi1	⊢ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) )
8	7	nic-idbl	⊢ ( ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) )
9	8	nic-imp	⊢ ( ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) ⊼ ( ( ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ) ⊼ ( ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ) ) )
10		nic-dfim	⊢ ( ( ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜓 → 𝜒 ) ) ⊼ ( ( ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ⊼ ( ( 𝜓 → 𝜒 ) ⊼ ( 𝜓 → 𝜒 ) ) ) )
11	10	nic-bi2	⊢ ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) )
12		nic-swap	⊢ ( ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ) )
13	11 12	nic-ich	⊢ ( ( 𝜓 → 𝜒 ) ⊼ ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ) )
14	13	nic-imp	⊢ ( ( ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ) ⊼ ( ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ) )
15	9 14	nic-ich	⊢ ( ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) ) ⊼ ( ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ) )
16	5 15	nic-ich	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ) )
17		nic-dfim	⊢ ( ( ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
18	17	nic-bi1	⊢ ( ( ( 𝜓 → 𝜒 ) ⊼ ( ( 𝜑 → 𝜒 ) ⊼ ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
19	16 18	nic-ich	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
20	2 19	nic-ich	⊢ ( ( 𝜑 → 𝜓 ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
21		nic-dfim	⊢ ( ( ( ( 𝜑 → 𝜓 ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ⊼ ( ( ( ( 𝜑 → 𝜓 ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ⊼ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) )
22	21	nic-bi1	⊢ ( ( ( 𝜑 → 𝜓 ) ⊼ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ⊼ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ⊼ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
23	20 22	nic-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )