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Theorem 3orel2 31920
 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.)
Assertion
Ref Expression
3orel2 𝜓 → ((𝜑𝜓𝜒) → (𝜑𝜒)))

Proof of Theorem 3orel2
StepHypRef Expression
1 3orrot 1077 . 2 ((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))
2 3orel1 1076 . . 3 𝜓 → ((𝜓𝜒𝜑) → (𝜒𝜑)))
3 orcom 401 . . 3 ((𝜒𝜑) ↔ (𝜑𝜒))
42, 3syl6ib 241 . 2 𝜓 → ((𝜓𝜒𝜑) → (𝜑𝜒)))
51, 4syl5bi 232 1 𝜓 → ((𝜑𝜓𝜒) → (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∨ 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-3or 1073 This theorem is referenced by:  nosep1o  32159  nosupbnd1lem5  32185
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