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	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$
	nic-ich.2	$$⊢\left(\psi ⊼\left(\chi ⊼\chi \right)\right)$$
Assertion	nic-ich	$$⊢\left(\phi ⊼\left(\chi ⊼\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nic-ich.1	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$
2		nic-ich.2	$$⊢\left(\psi ⊼\left(\chi ⊼\chi \right)\right)$$
3	2	nic-isw1	$$⊢\left(\left(\chi ⊼\chi \right)⊼\psi \right)$$
4	1	nic-imp	$$⊢\left(\left(\left(\chi ⊼\chi \right)⊼\psi \right)⊼\left(\left(\phi ⊼\left(\chi ⊼\chi \right)\right)⊼\left(\phi ⊼\left(\chi ⊼\chi \right)\right)\right)\right)$$
5	3 4	nic-mp	$$⊢\left(\phi ⊼\left(\chi ⊼\chi \right)\right)$$