ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.74 Unicode version

Theorem pm5.74 168
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 111 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
21imim3i 55 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ps )  -> 
( ph  ->  ch )
) )
3 bi2 121 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
43imim3i 55 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ch )  -> 
( ph  ->  ps )
) )
52, 4impbid 120 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
6 bi1 111 . . . 4  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
76pm2.86d 93 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
8 bi2 121 . . . 4  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  (
( ph  ->  ch )  ->  ( ph  ->  ps ) ) )
98pm2.86d 93 . . 3  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ch  ->  ps ) ) )
107, 9impbidd 118 . 2  |-  ( ( ( ph  ->  ps ) 
<->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  <->  ch )
) )
115, 10impbii 117 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  pm5.74i  169  pm5.74ri  170  pm5.74d  171  pm5.74rd  172  bibi2d  221
  Copyright terms: Public domain W3C validator