Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  orass Structured version   Visualization version   GIF version

Theorem orass 547
 Description: Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
orass (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Proof of Theorem orass
StepHypRef Expression
1 orcom 401 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑𝜓)))
2 or12 546 . 2 ((𝜒 ∨ (𝜑𝜓)) ↔ (𝜑 ∨ (𝜒𝜓)))
3 orcom 401 . . 3 ((𝜒𝜓) ↔ (𝜓𝜒))
43orbi2i 542 . 2 ((𝜑 ∨ (𝜒𝜓)) ↔ (𝜑 ∨ (𝜓𝜒)))
51, 2, 43bitri 286 1 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 382 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 This theorem is referenced by:  pm2.31  548  pm2.32  549  or32  550  or4  551  3orass  1075  axi12  2738  unass  3913  tppreqb  4481  ltxr  12142  lcmass  15529  plydivex  24251  clwwlkneq0  27156  disjxpin  29708  impor  34193  ifpim123g  38347
 Copyright terms: Public domain W3C validator