Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  andi3or Structured version   Visualization version   GIF version

Theorem andi3or 38820
 Description: Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)
Assertion
Ref Expression
andi3or ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)))

Proof of Theorem andi3or
StepHypRef Expression
1 andi 947 . . 3 ((𝜑 ∧ ((𝜓𝜒) ∨ 𝜃)) ↔ ((𝜑 ∧ (𝜓𝜒)) ∨ (𝜑𝜃)))
2 andi 947 . . . 4 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))
32orbi1i 543 . . 3 (((𝜑 ∧ (𝜓𝜒)) ∨ (𝜑𝜃)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (𝜑𝜃)))
41, 3bitri 264 . 2 ((𝜑 ∧ ((𝜓𝜒) ∨ 𝜃)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (𝜑𝜃)))
5 df-3or 1073 . . 3 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∨ 𝜃))
65anbi2i 732 . 2 ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ (𝜑 ∧ ((𝜓𝜒) ∨ 𝜃)))
7 df-3or 1073 . 2 (((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (𝜑𝜃)))
84, 6, 73bitr4i 292 1 ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∨ w3o 1071 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 384  df-an 385  df-3or 1073 This theorem is referenced by:  uneqsn  38821
 Copyright terms: Public domain W3C validator