Theorem nic-swap

Description: The connector -/\ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nic-swap	$$⊢\left(\left(\theta ⊼\phi \right)⊼\left(\left(\phi ⊼\theta \right)⊼\left(\phi ⊼\theta \right)\right)\right)$$

Proof

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