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Theorem pm5.32 640
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 296 . . . 4  |-  ( ( ps  <->  ch )  <->  ( -.  ps 
<->  -.  ch ) )
21imbi2i 313 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ph  ->  ( -.  ps  <->  -.  ch )
) )
3 pm5.74 247 . . 3  |-  ( (
ph  ->  ( -.  ps  <->  -. 
ch ) )  <->  ( ( ph  ->  -.  ps )  <->  (
ph  ->  -.  ch )
) )
4 notbi 296 . . 3  |-  ( ( ( ph  ->  -.  ps )  <->  ( ph  ->  -. 
ch ) )  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
52, 3, 43bitri 274 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
6 df-an 372 . . 3  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
7 df-an 372 . . 3  |-  ( (
ph  /\  ch )  <->  -.  ( ph  ->  -.  ch ) )
86, 7bibi12i 316 . 2  |-  ( ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) )  <->  ( -.  ( ph  ->  -.  ps )  <->  -.  ( ph  ->  -.  ch ) ) )
95, 8bitr4i 255 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
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:  pm5.32i  641  pm5.32d  643  xordi  903  rabbi  3004  rabxfrd  4642  asymref  5235  mpt22eqb  6419  cfilucfil4  22287  wl-ax11-lem8  31886  relexp0eq  36263  2sb5nd  36897  2sb5ndVD  37280  2sb5ndALT  37302
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