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Theorem nanim 1489
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 1488 . 2 ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑 → (𝜓𝜓)))
2 anidmdbi 569 . 2 ((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))
31, 2bitr2i 279 1 ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
Colors of variables: wff setvar class
Syntax hints:  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:  nic-dfim  1671  nic-ax  1675  waj-ax  33836  lukshef-ax2  33837
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