Theorem bj-bibibi

Description: A property of the biconditional. (Contributed by BJ, 26-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-bibibi	$$⊢\phi \leftrightarrow \left(\psi \leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.501	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)$$
2		bianir	$$⊢\left(\psi \wedge \left(\phi \leftrightarrow \psi \right)\right)\to \phi$$
3	2	ex	$$⊢\psi \to \left(\left(\phi \leftrightarrow \psi \right)\to \phi \right)$$
4		bibif	$$⊢\neg \psi \to \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \phi \right)$$
5	4	con2bid	$$⊢\neg \psi \to \left(\phi \leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)\right)$$
6	5	biimprd	$$⊢\neg \psi \to \left(\neg \left(\phi \leftrightarrow \psi \right)\to \phi \right)$$
7	3 6	bija	$$⊢\left(\psi \leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)\to \phi$$
8	1 7	impbii	$$⊢\phi \leftrightarrow \left(\psi \leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)$$