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	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		exmid	⊢ ( 𝜑 ∨ ¬ 𝜑 )
2		ordi	⊢ ( ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∨ ¬ 𝜑 ) ∧ ( 𝜑 ∨ 𝜓 ) ) )
3	1 2	mpbiran	⊢ ( ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∨ 𝜓 ) )
4	3	bicomi	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) )