Theorem aibandbiaiffaiffb

Description: A closed form showing (a implies b and b implies a) same-as (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016)

	Ref	Expression
Assertion	aibandbiaiffaiffb	$$⊢\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \phi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dfbi2	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\psi \to \phi \right)\right)$$
2	1	bicomi	$$⊢\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \phi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$