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Theorem nan 580
Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
Assertion
Ref Expression
nan  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  ( ( ph  /\  ps )  ->  -.  ch ) )

Proof of Theorem nan
StepHypRef Expression
1 impexp 446 . 2  |-  ( ( ( ph  /\  ps )  ->  -.  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
2 imnan 422 . . 3  |-  ( ( ps  ->  -.  ch )  <->  -.  ( ps  /\  ch ) )
32imbi2i 312 . 2  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  ( ph  ->  -.  ( ps  /\  ch ) ) )
41, 3bitr2i 250 1  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  ( ( ph  /\  ps )  ->  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369
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 185  df-an 371
This theorem is referenced by:  pm4.15  581  somincom  5344  wemaplem2  7873  alephval3  8392  hauspwpwf1  19693  stoweidlem34  29978  stirlinglem5  30022
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