Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3ornot23 Structured version   Visualization version   GIF version

Theorem 3ornot23 41213
 Description: If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 41551. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
3ornot23 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))

Proof of Theorem 3ornot23
StepHypRef Expression
1 idd 24 . . 3 𝜑 → (𝜒𝜒))
2 pm2.21 123 . . 3 𝜑 → (𝜑𝜒))
3 pm2.21 123 . . 3 𝜓 → (𝜓𝜒))
41, 2, 33jaao 1430 . 2 ((¬ 𝜑 ∧ ¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
543anidm12 1416 1 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ w3o 1083 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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086 This theorem is referenced by:  tratrb  41240  tratrbVD  41565
 Copyright terms: Public domain W3C validator