Theorem biimpr

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999) (Proof shortened by Wolf Lammen, 11-Nov-2012)

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

Proof

Step	Hyp	Ref	Expression
1		dfbi1	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\left(\phi \to \psi \right)\to \neg \left(\psi \to \phi \right)\right)$$
2		simprim	$$⊢\neg \left(\left(\phi \to \psi \right)\to \neg \left(\psi \to \phi \right)\right)\to \left(\psi \to \phi \right)$$
3	1 2	sylbi	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$