Theorem ordi 908

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Jan-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 907	. 2 ⊢ ((¬ 𝜑 → (𝜓 ∧ 𝜒)) ↔ ((¬ 𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
2		df-or 385	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ (¬ 𝜑 → (𝜓 ∧ 𝜒)))
3		df-or 385	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
4		df-or 385	. . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (¬ 𝜑 → 𝜒))
5	3, 4	anbi12i 733	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ 𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
6	1, 2, 5	3bitr4i 292	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  ordir  909  orddi  913  pm5.63  959  pm4.43  968  cadan  1548  undi  3874  undif3  3888  undif4  4035  elnn1uz2  11765  or3di  29307  ifpan23  37804  ifpidg  37836  ifpim123g  37845