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Theorem pm5.74 237
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 bi1 180 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
21imim3i 57 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ps )  -> 
( ph  ->  ch )
) )
3 bi2 191 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
43imim3i 57 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ch )  -> 
( ph  ->  ps )
) )
52, 4impbid 185 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
6 bi1 180 . . . 4  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
76pm2.86d 95 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
8 bi2 191 . . . 4  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  (
( ph  ->  ch )  ->  ( ph  ->  ps ) ) )
98pm2.86d 95 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ch  ->  ps ) ) )
107, 9impbidd 183 . 2  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  <->  ch )
) )
115, 10impbii 182 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178
This theorem is referenced by:  pm5.74i  238  pm5.74ri  239  pm5.74d  240  pm5.74rd  241  bibi2d  311  pm5.32  620  orbidi  903
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
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