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Theorem pm5.74 252
Description: Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.)
Assertion
Ref Expression
pm5.74  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )

Proof of Theorem pm5.74
StepHypRef Expression
1 biimp 198 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
21imim3i 61 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ps )  -> 
( ph  ->  ch )
) )
3 biimpr 203 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
43imim3i 61 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ch )  -> 
( ph  ->  ps )
) )
52, 4impbid 195 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
6 biimp 198 . . . 4  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
76pm2.86d 102 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
8 biimpr 203 . . . 4  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  (
( ph  ->  ch )  ->  ( ph  ->  ps ) ) )
98pm2.86d 102 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ch  ->  ps ) ) )
107, 9impbidd 193 . 2  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  <->  ch )
) )
115, 10impbii 192 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189
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 190
This theorem is referenced by:  pm5.74i  253  pm5.74ri  254  pm5.74d  255  pm5.74rd  256  bibi2d  324  pm5.32  646  orbidi  948
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