Theorem nic-bi2

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

	Ref	Expression
Hypothesis	nic-bi2.1	$$⊢\left(\left(\phi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\psi ⊼\psi \right)\right)\right)$$
Assertion	nic-bi2	$$⊢\left(\psi ⊼\left(\phi ⊼\phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nic-bi2.1	$$⊢\left(\left(\phi ⊼\psi \right)⊼\left(\left(\phi ⊼\phi \right)⊼\left(\psi ⊼\psi \right)\right)\right)$$
2	1	nic-isw2	$$⊢\left(\left(\phi ⊼\psi \right)⊼\left(\left(\psi ⊼\psi \right)⊼\left(\phi ⊼\phi \right)\right)\right)$$
3		nic-id	$$⊢\left(\psi ⊼\left(\psi ⊼\psi \right)\right)$$
4	2 3	nic-iimp1	$$⊢\left(\psi ⊼\left(\phi ⊼\psi \right)\right)$$
5	4	nic-idel	$$⊢\left(\psi ⊼\left(\phi ⊼\phi \right)\right)$$