Theorem pm5.63 890

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 404	. . 3 ⊢ (φ ∨ ¬ φ)
2		ordi 834	. . 3 ⊢ ((φ ∨ (¬ φ ∧ ψ)) ↔ ((φ ∨ ¬ φ) ∧ (φ ∨ ψ)))
3	1, 2	mpbiran 884	. 2 ⊢ ((φ ∨ (¬ φ ∧ ψ)) ↔ (φ ∨ ψ))
4	3	bicomi 193	1 ⊢ ((φ ∨ ψ) ↔ (φ ∨ (¬ φ ∧ ψ)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358

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

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  cad1  1398