Theorem nic-idlem1

Description: Lemma for nic-id . (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nic-idlem1	⊢ ( ( 𝜃 ⊼ ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ 𝜃 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nic-ax	⊢ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ ( ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ⊼ ( ( 𝜑 ⊼ 𝜒 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ) ) )
2	1	nic-imp	⊢ ( ( 𝜃 ⊼ ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ 𝜃 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ 𝜃 ) ) )