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Theorem df3an2 37887
Description: Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
Assertion
Ref Expression
df3an2 ((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))

Proof of Theorem df3an2
StepHypRef Expression
1 df-3an 1039 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 df-an 386 . . 3 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜑𝜓) → ¬ 𝜒))
3 impexp 462 . . 3 (((𝜑𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
42, 3xchbinx 324 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
51, 4bitri 264 1 ((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037
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 386  df-3an 1039
This theorem is referenced by: (None)
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