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Theorem nannan 1619
 Description: Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)
Assertion
Ref Expression
nannan ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Proof of Theorem nannan
StepHypRef Expression
1 nanimn 1615 . 2 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
2 nanan 1614 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 328 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
41, 3bitr4i 270 1 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ⊼ wnan 1612 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 199  df-an 387  df-nan 1613 This theorem is referenced by:  nanim  1621  nanbi  1624  nanass  1635  nanassOLD  1636  nic-mp  1770  nic-ax  1772  waj-ax  32935  lukshef-ax2  32936  arg-ax  32937
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