Theorem bj-dfbi5

Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019)

	Ref	Expression
Assertion	bj-dfbi5	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\to \left(\phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orcom	$$⊢\left(\left(\phi \wedge \psi \right)\vee \neg \left(\phi \vee \psi \right)\right)\leftrightarrow \left(\neg \left(\phi \vee \psi \right)\vee \left(\phi \wedge \psi \right)\right)$$
2		bj-dfbi4	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\vee \neg \left(\phi \vee \psi \right)\right)$$
3		imor	$$⊢\left(\left(\phi \vee \psi \right)\to \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\neg \left(\phi \vee \psi \right)\vee \left(\phi \wedge \psi \right)\right)$$
4	1 2 3	3bitr4i	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\to \left(\phi \wedge \psi \right)\right)$$