Theorem pm5.55

Description: Theorem *5.55 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 20-Jan-2013)

	Ref	Expression
Assertion	pm5.55	$$⊢\left(\left(\left(\phi \vee \psi \right)\leftrightarrow \phi \right)\vee \left(\left(\phi \vee \psi \right)\leftrightarrow \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		biort	$$⊢\phi \to \left(\phi \leftrightarrow \left(\phi \vee \psi \right)\right)$$
2	1	bicomd	$$⊢\phi \to \left(\left(\phi \vee \psi \right)\leftrightarrow \phi \right)$$
3		biorf	$$⊢\neg \phi \to \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$
4	3	bicomd	$$⊢\neg \phi \to \left(\left(\phi \vee \psi \right)\leftrightarrow \psi \right)$$
5	2 4	nsyl5	$$⊢\neg \left(\left(\phi \vee \psi \right)\leftrightarrow \phi \right)\to \left(\left(\phi \vee \psi \right)\leftrightarrow \psi \right)$$
6	5	orri	$$⊢\left(\left(\left(\phi \vee \psi \right)\leftrightarrow \phi \right)\vee \left(\left(\phi \vee \psi \right)\leftrightarrow \psi \right)\right)$$