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Theorem nannan 1491
 Description: Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
Assertion
Ref Expression
nannan ((𝜑 ⊼ (𝜒𝜓)) ↔ (𝜑 → (𝜒𝜓)))

Proof of Theorem nannan
StepHypRef Expression
1 imnan 437 . 2 ((𝜑 → ¬ (𝜒𝜓)) ↔ ¬ (𝜑 ∧ (𝜒𝜓)))
2 nanan 1489 . . 3 ((𝜒𝜓) ↔ ¬ (𝜒𝜓))
32imbi2i 325 . 2 ((𝜑 → (𝜒𝜓)) ↔ (𝜑 → ¬ (𝜒𝜓)))
4 df-nan 1488 . 2 ((𝜑 ⊼ (𝜒𝜓)) ↔ ¬ (𝜑 ∧ (𝜒𝜓)))
51, 3, 43bitr4ri 293 1 ((𝜑 ⊼ (𝜒𝜓)) ↔ (𝜑 → (𝜒𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ⊼ wnan 1487 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 385  df-nan 1488 This theorem is referenced by:  nanim  1492  nanbi  1494  nic-mp  1636  nic-ax  1638  waj-ax  32538  lukshef-ax2  32539  arg-ax  32540  rp-fakenanass  38177
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