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Theorem nanim 1482
Description: Implication in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.)
Assertion
Ref Expression
nanim ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))

Proof of Theorem nanim
StepHypRef Expression
1 nannan 1481 . 2 ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑 → (𝜓𝜓)))
2 anidmdbi 566 . 2 ((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))
31, 2bitr2i 277 1 ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wnan 1475
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 208  df-an 397  df-nan 1476
This theorem is referenced by:  nic-dfim  1661  nic-ax  1665  waj-ax  33659  lukshef-ax2  33660
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