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

Theorem 3ori 1534
Description: Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
Hypothesis
Ref Expression
3ori.1 (𝜑𝜓𝜒)
Assertion
Ref Expression
3ori ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)

Proof of Theorem 3ori
StepHypRef Expression
1 ioran 912 . 2 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2 3ori.1 . . . 4 (𝜑𝜓𝜒)
3 df-3or 1071 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
42, 3mpbi 220 . . 3 ((𝜑𝜓) ∨ 𝜒)
54ori 841 . 2 (¬ (𝜑𝜓) → 𝜒)
61, 5sylbir 225 1 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 826  w3o 1069
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-an 383  df-or 827  df-3or 1071
This theorem is referenced by:  rankxplim3  8907
  Copyright terms: Public domain W3C validator