Theorem pm4.43

Description: Theorem *4.43 of WhiteheadRussell p. 119. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 26-Nov-2012)

	Ref	Expression
Assertion	pm4.43	$$⊢\phi \leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \left(\phi \vee \neg \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm3.24	$$⊢\neg \left(\psi \wedge \neg \psi \right)$$
2	1	biorfi	$$⊢\phi \leftrightarrow \left(\phi \vee \left(\psi \wedge \neg \psi \right)\right)$$
3		ordi	$$⊢\left(\phi \vee \left(\psi \wedge \neg \psi \right)\right)\leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \left(\phi \vee \neg \psi \right)\right)$$
4	2 3	bitri	$$⊢\phi \leftrightarrow \left(\left(\phi \vee \psi \right)\wedge \left(\phi \vee \neg \psi \right)\right)$$