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Theorem nan 582
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 447 . 2  |-  ( ( ( ph  /\  ps )  ->  -.  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
2 imnan 423 . . 3  |-  ( ( ps  ->  -.  ch )  <->  -.  ( ps  /\  ch ) )
32imbi2i 313 . 2  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  ( ph  ->  -.  ( ps  /\  ch ) ) )
41, 3bitr2i 253 1  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  ( ( ph  /\  ps )  ->  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370
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 188  df-an 372
This theorem is referenced by:  pm4.15  583  somincom  5245  wemaplem2  8053  alephval3  8530  hauspwpwf1  20939  icccncfext  37385  stoweidlem34  37512  stirlinglem5  37557  fourierdlem42  37628  etransc  37763
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