Theorem nic-luk2

Description: Proof of luk-2 from nic-ax and nic-mp . (Contributed by Jeff Hoffman, 18-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nic-luk2	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nic-dfim	⊢ ( ( ( ¬ 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 → 𝜑 ) ) ⊼ ( ( ( ¬ 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) ) ⊼ ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( ¬ 𝜑 → 𝜑 ) ) ) )
2	1	nic-bi2	⊢ ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( ( ¬ 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) ) )
3		nic-dfneg	⊢ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ¬ 𝜑 ) ⊼ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 ⊼ ¬ 𝜑 ) ) )
4		nic-id	⊢ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) )
5	3 4	nic-iimp1	⊢ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ¬ 𝜑 ) )
6	5	nic-isw2	⊢ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( ¬ 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) )
7	2 6	nic-iimp1	⊢ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( ¬ 𝜑 → 𝜑 ) )
8	7	nic-isw1	⊢ ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) )
9		nic-dfim	⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) ⊼ ( ( ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ) ⊼ ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ⊼ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) ) )
10	9	nic-bi1	⊢ ( ( ( ¬ 𝜑 → 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ⊼ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) )
11	8 10	nic-mp	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )