Theorem nic-ich

Description: Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nic-ich.1	⊢ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) )
	nic-ich.2	⊢ ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) )
Assertion	nic-ich	⊢ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		nic-ich.1	⊢ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) )
2		nic-ich.2	⊢ ( 𝜓 ⊼ ( 𝜒 ⊼ 𝜒 ) )
3	2	nic-isw1	⊢ ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 )
4	1	nic-imp	⊢ ( ( ( 𝜒 ⊼ 𝜒 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ⊼ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) ) ) )
5	3 4	nic-mp	⊢ ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜒 ) )