Theorem nic-bi1

Description: Inference to extract one side of an implication from a definition. (Contributed by Jeff Hoffman, 18-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nic-bi1.1	$$⊢\left(\left(\phi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\psi ⊼\psi \right)\right)\right)$$
Assertion	nic-bi1	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nic-bi1.1	$$⊢\left(\left(\phi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\psi ⊼\psi \right)\right)\right)$$
2		nic-id	$$⊢\left(\phi ⊼\left(\phi ⊼\phi \right)\right)$$
3	1 2	nic-iimp1	$$⊢\left(\phi ⊼\left(\phi ⊼\psi \right)\right)$$
4	3	nic-isw2	$$⊢\left(\phi ⊼\left(\psi ⊼\phi \right)\right)$$
5	4	nic-idel	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$