Theorem nic-idbl 1689

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.)

Hypothesis

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

Assertion

Ref	Expression
nic-idbl	⊢ ((𝜓 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)))

Proof of Theorem nic-idbl

Step	Hyp	Ref	Expression
1		nic-idbl.1	. . 3 ⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓))
2	1	nic-imp 1678	. 2 ⊢ ((𝜓 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜓) ⊼ (𝜑 ⊼ 𝜓)))
3	1	nic-imp 1678	. 2 ⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)))
4	2, 3	nic-ich 1688	1 ⊢ ((𝜓 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)))

Syntax hints:   ⊼ wnan 1486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-nan 1487

This theorem is referenced by:  nic-luk1  1694