Theorem ibibr

Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005) (Proof shortened by Wolf Lammen, 21-Dec-2013)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.501	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)$$
2		bicom	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\psi \leftrightarrow \phi \right)$$
3	1 2	syl6bb	$$⊢\phi \to \left(\psi \leftrightarrow \left(\psi \leftrightarrow \phi \right)\right)$$
4	3	pm5.74i	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \left(\psi \leftrightarrow \phi \right)\right)$$