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Theorem pm5.32 620
Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
pm5.32  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )

Proof of Theorem pm5.32
StepHypRef Expression
1 notbi 288 . . . 4  |-  ( ( ps  <->  ch )  <->  ( -.  ps 
<->  -.  ch ) )
21imbi2i 305 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ph  ->  ( -.  ps  <->  -.  ch )
) )
3 pm5.74 237 . . 3  |-  ( (
ph  ->  ( -.  ps  <->  -. 
ch ) )  <->  ( ( ph  ->  -.  ps )  <->  (
ph  ->  -.  ch )
) )
4 notbi 288 . . 3  |-  ( ( ( ph  ->  -.  ps )  <->  ( ph  ->  -. 
ch ) )  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
52, 3, 43bitri 264 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
6 df-an 362 . . 3  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
7 df-an 362 . . 3  |-  ( (
ph  /\  ch )  <->  -.  ( ph  ->  -.  ch ) )
86, 7bibi12i 308 . 2  |-  ( ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) )  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
95, 8bitr4i 245 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360
This theorem is referenced by:  pm5.32i  621  pm5.32d  623  xordi  870  cbval2  2048  cbvex2  2049  rabbi  2693  rabxfrd  4527  asymref  5047  mpt22eqb  5887  dvdslelem  12536  2sb5nd  27462  2sb5ndVD  27819  2sb5ndALT  27842
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-an 362
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