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Theorem nannan 1488
 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 1485 . 2 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
2 nanan 1484 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 339 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
41, 3bitr4i 281 1 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ⊼ wnan 1482 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-nan 1483 This theorem is referenced by:  nanim  1489  nanbi  1491  nanass  1501  nic-mp  1673  nic-ax  1675  waj-ax  33836  lukshef-ax2  33837  arg-ax  33838
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