Theorem nic-luk3

Description: Proof of luk-3 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-luk3	⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) )

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-bi1	⊢ ( ( ¬ 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( ( ¬ 𝜑 → 𝜓 ) ⊼ ( ¬ 𝜑 → 𝜓 ) ) )
3		nic-dfneg	⊢ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ¬ 𝜑 ) ⊼ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 ⊼ ¬ 𝜑 ) ) )
4	3	nic-bi2	⊢ ( ¬ 𝜑 ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) )
5		nic-id	⊢ ( 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) )
6	4 5	nic-iimp1	⊢ ( 𝜑 ⊼ ¬ 𝜑 )
7	2 6	nic-iimp2	⊢ ( 𝜑 ⊼ ( ( ¬ 𝜑 → 𝜓 ) ⊼ ( ¬ 𝜑 → 𝜓 ) ) )
8		nic-dfim	⊢ ( ( ( 𝜑 ⊼ ( ( ¬ 𝜑 → 𝜓 ) ⊼ ( ¬ 𝜑 → 𝜓 ) ) ) ⊼ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ) ⊼ ( ( ( 𝜑 ⊼ ( ( ¬ 𝜑 → 𝜓 ) ⊼ ( ¬ 𝜑 → 𝜓 ) ) ) ⊼ ( 𝜑 ⊼ ( ( ¬ 𝜑 → 𝜓 ) ⊼ ( ¬ 𝜑 → 𝜓 ) ) ) ) ⊼ ( ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ⊼ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ) ) )
9	8	nic-bi1	⊢ ( ( 𝜑 ⊼ ( ( ¬ 𝜑 → 𝜓 ) ⊼ ( ¬ 𝜑 → 𝜓 ) ) ) ⊼ ( ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ⊼ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ) )
10	7 9	nic-mp	⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) )