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Theorem nanimn 1612
Description: Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.)
Assertion
Ref Expression
nanimn ((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))

Proof of Theorem nanimn
StepHypRef Expression
1 df-nan 1610 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 imnan 389 . 2 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
31, 2bitr4i 270 1 ((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wnan 1609
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 199  df-an 386  df-nan 1610
This theorem is referenced by:  nancom  1614  nannan  1616  nannot  1619  nanbi1  1622
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