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Theorem pm5.32 617
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 286 . . . 4  |-  ( ( ps  <->  ch )  <->  ( -.  ps 
<->  -.  ch ) )
21imbi2i 303 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ph  ->  ( -.  ps  <->  -.  ch )
) )
3 pm5.74 235 . . 3  |-  ( (
ph  ->  ( -.  ps  <->  -. 
ch ) )  <->  ( ( ph  ->  -.  ps )  <->  (
ph  ->  -.  ch )
) )
4 notbi 286 . . 3  |-  ( ( ( ph  ->  -.  ps )  <->  ( ph  ->  -. 
ch ) )  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
52, 3, 43bitri 262 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
6 df-an 360 . . 3  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
7 df-an 360 . . 3  |-  ( (
ph  /\  ch )  <->  -.  ( ph  ->  -.  ch ) )
86, 7bibi12i 306 . 2  |-  ( ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) )  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
95, 8bitr4i 243 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  pm5.32i  618  pm5.32d  620  xordi  865  cbval2  1946  cbvex2  1947  rabbi  2720  rabxfrd  4557  asymref  5061  mpt22eqb  5955  dvdslelem  12575  2sb5nd  28382  2sb5ndVD  28759  2sb5ndALT  28782
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 177  df-an 360
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