Theorem nic-isw2

Description: Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nic-isw2.1	$$⊢\left(\psi ⊼\left(\theta ⊼\phi \right)\right)$$
Assertion	nic-isw2	$$⊢\left(\psi ⊼\left(\phi ⊼\theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nic-isw2.1	$$⊢\left(\psi ⊼\left(\theta ⊼\phi \right)\right)$$
2		nic-swap	$$⊢\left(\left(\phi ⊼\theta \right)⊼\left(\left(\theta ⊼\phi \right)⊼\left(\theta ⊼\phi \right)\right)\right)$$
3	2	nic-imp	$$⊢\left(\left(\psi ⊼\left(\theta ⊼\phi \right)\right)⊼\left(\left(\left(\phi ⊼\theta \right)⊼\psi \right)⊼\left(\left(\phi ⊼\theta \right)⊼\psi \right)\right)\right)$$
4	1 3	nic-mp	$$⊢\left(\left(\phi ⊼\theta \right)⊼\psi \right)$$
5	4	nic-isw1	$$⊢\left(\psi ⊼\left(\phi ⊼\theta \right)\right)$$