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	$$⊢\left(\phi ⊼\left(\chi ⊼\psi \right)\right)$$
Assertion	nic-imp	$$⊢\left(\left(\theta ⊼\chi \right)⊼\left(\left(\phi ⊼\theta \right)⊼\left(\phi ⊼\theta \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nic-imp.1	$$⊢\left(\phi ⊼\left(\chi ⊼\psi \right)\right)$$
2		nic-ax	$$⊢\left(\left(\phi ⊼\left(\chi ⊼\psi \right)\right)⊼\left(\left(\tau ⊼\left(\tau ⊼\tau \right)\right)⊼\left(\left(\theta ⊼\chi \right)⊼\left(\left(\phi ⊼\theta \right)⊼\left(\phi ⊼\theta \right)\right)\right)\right)\right)$$
3	1 2	nic-mp	$$⊢\left(\left(\theta ⊼\chi \right)⊼\left(\left(\phi ⊼\theta \right)⊼\left(\phi ⊼\theta \right)\right)\right)$$