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Theorem nannan 1448
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 438 . 2 ((𝜑 → ¬ (𝜒𝜓)) ↔ ¬ (𝜑 ∧ (𝜒𝜓)))
2 nanan 1446 . . 3 ((𝜒𝜓) ↔ ¬ (𝜒𝜓))
32imbi2i 326 . 2 ((𝜑 → (𝜒𝜓)) ↔ (𝜑 → ¬ (𝜒𝜓)))
4 df-nan 1445 . 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 384  wnan 1444
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-nan 1445
This theorem is referenced by:  nanim  1449  nanbi  1451  nic-mp  1593  nic-ax  1595  waj-ax  32108  lukshef-ax2  32109  arg-ax  32110  rp-fakenanass  37380
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