Theorem pm5.63 1037

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 908	. . 3 ⊢ (𝜑 ∨ ¬ 𝜑)
2		ordi 1023	. . 3 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜓)))
3	1, 2	mpbiran 722	. 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) ↔ (𝜑 ∨ 𝜓))
4	3	bicomi 227	1 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  jaoi2  1075  plydivex  26618  lineunray  36912  wl-df4-3mintru2  38410