Theorem pm5.63 1016

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

Assertion

Ref	Expression
pm5.63	⊢ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))

Proof of Theorem pm5.63

Step	Hyp	Ref	Expression
1		exmid 891	. . 3 ⊢ (𝜑 ∨ ¬ 𝜑)
2		ordi 1002	. . 3 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜓)))
3	1, 2	mpbiran 705	. 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) ↔ (𝜑 ∨ 𝜓))
4	3	bicomi 223	1 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844

This theorem is referenced by:  jaoi2  1056  plydivex  25362  lineunray  34376  wl-df4-3mintru2  35585