Theorem nic-imp

Description: Inference for nic-mp using nic-ax as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nic-imp.1	⊢ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) )
Assertion	nic-imp	⊢ ( ( 𝜃 ⊼ 𝜒 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nic-imp.1	⊢ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) )
2		nic-ax	⊢ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ ( ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ⊼ ( ( 𝜃 ⊼ 𝜒 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) )
3	1 2	nic-mp	⊢ ( ( 𝜃 ⊼ 𝜒 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) )