MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3orel2 Structured version   Visualization version   GIF version

Theorem 3orel2 1483
Description: Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Eric Schmidt, 8-Oct-2025.)
Assertion
Ref Expression
3orel2 𝜓 → ((𝜑𝜓𝜒) → (𝜑𝜒)))

Proof of Theorem 3orel2
StepHypRef Expression
1 3orcoma 1092 . 2 ((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))
2 3orel1 1090 . 2 𝜓 → ((𝜓𝜑𝜒) → (𝜑𝜒)))
31, 2biimtrid 242 1 𝜓 → ((𝜑𝜓𝜒) → (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 847  w3o 1085
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 207  df-or 848  df-3or 1087
This theorem is referenced by:  nogesgn1o  27733  nosep1o  27741  nosupbnd1lem5  27772
  Copyright terms: Public domain W3C validator