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	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$
Assertion	nic-idbl	$$⊢\left(\left(\psi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\phi ⊼\phi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nic-idbl.1	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$
2	1	nic-imp	$$⊢\left(\left(\psi ⊼\psi \right)⊼\left(\left(\phi ⊼\psi \right)⊼\left(\phi ⊼\psi \right)\right)\right)$$
3	1	nic-imp	$$⊢\left(\left(\phi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\phi ⊼\phi \right)\right)\right)$$
4	2 3	nic-ich	$$⊢\left(\left(\psi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\phi ⊼\phi \right)\right)\right)$$