Theorem impbii

Description: Infer an equivalence from an implication and its converse. Inference associated with impbi . (Contributed by NM, 29-Dec-1992)

	Ref	Expression
Hypotheses	impbii.1	$$⊢\phi \to \psi$$
	impbii.2	$$⊢\psi \to \phi$$
Assertion	impbii	$$⊢\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		impbii.1	$$⊢\phi \to \psi$$
2		impbii.2	$$⊢\psi \to \phi$$
3		impbi	$$⊢\left(\phi \to \psi \right)\to \left(\left(\psi \to \phi \right)\to \left(\phi \leftrightarrow \psi \right)\right)$$
4	1 2 3	mp2	$$⊢\phi \leftrightarrow \psi$$