Theorem pm5.63

Description: Theorem *5.63 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 25-Dec-2012)

	Ref	Expression
Assertion	pm5.63	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \left(\neg \phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		exmid	$$⊢\left(\phi \vee \neg \phi \right)$$
2		ordi	$$⊢\left(\phi \vee \left(\neg \phi \wedge \psi \right)\right)\leftrightarrow \left(\left(\phi \vee \neg \phi \right)\wedge \left(\phi \vee \psi \right)\right)$$
3	1 2	mpbiran	$$⊢\left(\phi \vee \left(\neg \phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \vee \psi \right)$$
4	3	bicomi	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \left(\neg \phi \wedge \psi \right)\right)$$