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Theorem nan 564
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 434 . 2  |-  ( ( ( ph  /\  ps )  ->  -.  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
2 imnan 412 . . 3  |-  ( ( ps  ->  -.  ch )  <->  -.  ( ps  /\  ch ) )
32imbi2i 304 . 2  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  ( ph  ->  -.  ( ps  /\  ch ) ) )
41, 3bitr2i 242 1  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  ( ( ph  /\  ps )  ->  -.  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359
This theorem is referenced by:  pm4.15  565  somincom  5263  wemaplem2  7508  alephval3  7983  hauspwpwf1  18011  stoweidlem34  27750  stirlinglem5  27794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-an 361
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