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Theorem df3or2 38580
 Description: Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
Assertion
Ref Expression
df3or2 ((𝜑𝜓𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))

Proof of Theorem df3or2
StepHypRef Expression
1 df-3or 1073 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
2 df-or 384 . . 3 (((𝜑𝜓) ∨ 𝜒) ↔ (¬ (𝜑𝜓) → 𝜒))
3 ioran 512 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
43imbi1i 338 . . . 4 ((¬ (𝜑𝜓) → 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
5 impexp 461 . . . 4 (((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))
64, 5bitri 264 . . 3 ((¬ (𝜑𝜓) → 𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))
72, 6bitri 264 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))
81, 7bitri 264 1 ((𝜑𝜓𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∨ 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-an 385  df-3or 1073 This theorem is referenced by: (None)
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