Theorem nic-idbl

Description: Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nic-idbl.1	⊢ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) )
Assertion	nic-idbl	⊢ ( ( 𝜓 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nic-idbl.1	⊢ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) )
2	1	nic-imp	⊢ ( ( 𝜓 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜓 ) ⊼ ( 𝜑 ⊼ 𝜓 ) ) )
3	1	nic-imp	⊢ ( ( 𝜑 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) )
4	2 3	nic-ich	⊢ ( ( 𝜓 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) )