Theorem ordi 834

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)

Assertion

Ref	Expression
ordi	⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ)))

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		jcab 833	. 2 ⊢ ((¬ φ → (ψ ∧ χ)) ↔ ((¬ φ → ψ) ∧ (¬ φ → χ)))
2		df-or 359	. 2 ⊢ ((φ ∨ (ψ ∧ χ)) ↔ (¬ φ → (ψ ∧ χ)))
3		df-or 359	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
4		df-or 359	. . 3 ⊢ ((φ ∨ χ) ↔ (¬ φ → χ))
5	3, 4	anbi12i 678	. 2 ⊢ (((φ ∨ ψ) ∧ (φ ∨ χ)) ↔ ((¬ φ → ψ) ∧ (¬ φ → χ)))
6	1, 2, 5	3bitr4i 268	1 ⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ 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:  ordir  835  orddi  839  pm5.63  890  pm4.43  893  cadan  1392  undi  3503  undif4  3608