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Theorem nan 828
 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 454 . 2 (((𝜑𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2 imnan 403 . . 3 ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 339 . 2 ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
41, 3bitr2i 279 1 ((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399 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 This theorem is referenced by:  pm4.15  831  somincom  5971  wemaplem2  9057  alephval3  9583  hauspwpwf1  22700  rtprmirr  39879  icccncfext  42930  stoweidlem34  43077  stirlinglem5  43121  fourierdlem42  43192  etransc  43326
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