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Theorem 3ori 1236
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 704 . 2 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2 3ori.1 . . . 4 (𝜑𝜓𝜒)
3 df-3or 925 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
42, 3mpbi 143 . . 3 ((𝜑𝜓) ∨ 𝜒)
54ori 677 . 2 (¬ (𝜑𝜓) → 𝜒)
61, 5sylbir 133 1 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  w3o 923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-3or 925
This theorem is referenced by: (None)
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