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Theorem df3nandALT2 33808
Description: The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
Assertion
Ref Expression
df3nandALT2 ((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))

Proof of Theorem df3nandALT2
StepHypRef Expression
1 df-3nand 33806 . 2 ((𝜑𝜓𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2 imnan 403 . . 3 ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 339 . 2 ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
4 imnan 403 . . 3 ((𝜑 → ¬ (𝜓𝜒)) ↔ ¬ (𝜑 ∧ (𝜓𝜒)))
5 3anass 1092 . . 3 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
64, 5xchbinxr 338 . 2 ((𝜑 → ¬ (𝜓𝜒)) ↔ ¬ (𝜑𝜓𝜒))
71, 3, 63bitri 300 1 ((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084  w3nand 33805
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 210  df-an 400  df-3an 1086  df-3nand 33806
This theorem is referenced by: (None)
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