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Theorem nan 830
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 450 . 2 (((𝜑𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2 imnan 399 . . 3 ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 336 . 2 ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
41, 3bitr2i 276 1 ((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395
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 207  df-an 396
This theorem is referenced by:  pm4.15  833  somincom  6157  wemaplem2  9585  alephval3  10148  hauspwpwf1  24011  rtprmirr  26818  sticksstones22  42150  icccncfext  45843  stoweidlem34  45990  stirlinglem5  46034  fourierdlem42  46105  etransc  46239
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