Theorem nanim

Description: Implication in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007)

	Ref	Expression
Assertion	nanim	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nannan	$$⊢\left(\phi ⊼\left(\psi ⊼\psi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \wedge \psi \right)\right)$$
2		anidmdbi	$$⊢\left(\phi \to \left(\psi \wedge \psi \right)\right)\leftrightarrow \left(\phi \to \psi \right)$$
3	1 2	bitr2i	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi ⊼\left(\psi ⊼\psi \right)\right)$$