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Theorem nan 602
 Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
Assertion
Ref Expression
nan ((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))

Proof of Theorem nan
StepHypRef Expression
1 impexp 461 . 2 (((𝜑𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2 imnan 437 . . 3 ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 325 . 2 ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
41, 3bitr2i 264 1 ((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383 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 196  df-an 385 This theorem is referenced by:  pm4.15  603  somincom  5449  wemaplem2  8335  alephval3  8816  hauspwpwf1  21601  icccncfext  38773  stoweidlem34  38927  stirlinglem5  38971  fourierdlem42  39042  etransc  39176
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