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Theorem pm5.32 434
Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
Assertion
Ref Expression
pm5.32  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )

Proof of Theorem pm5.32
StepHypRef Expression
1 id 19 . . 3  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ph  ->  ( ps  <->  ch )
) )
21pm5.32d 431 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  /\  ps )  <->  ( ph  /\ 
ch ) ) )
3 ibar 289 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
4 ibar 289 . . . 4  |-  ( ph  ->  ( ch  <->  ( ph  /\ 
ch ) ) )
53, 4bibi12d 228 . . 3  |-  ( ph  ->  ( ( ps  <->  ch )  <->  ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) ) ) )
65biimprcd 153 . 2  |-  ( ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) )  ->  ( ph  ->  ( ps  <->  ch )
) )
72, 6impbii 121 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  pm5.32i  435  xordidc  1306  cbvex2  1813  rabbi  2504  rabxfrd  4229  asymref  4738  rexrnmpt  5338  mpt22eqb  5638
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