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Theorem pm5.32 450
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 447 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  ->  ( ( ph  /\  ps )  <->  ( ph  /\ 
ch ) ) )
3 ibar 299 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
4 ibar 299 . . . 4  |-  ( ph  ->  ( ch  <->  ( ph  /\ 
ch ) ) )
53, 4bibi12d 234 . . 3  |-  ( ph  ->  ( ( ps  <->  ch )  <->  ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) ) ) )
65biimprcd 159 . 2  |-  ( ( ( ph  /\  ps ) 
<->  ( ph  /\  ch ) )  ->  ( ph  ->  ( ps  <->  ch )
) )
72, 6impbii 125 1  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  pm5.32i  451  biadani  607  xordidc  1394  cbvex2  1915  rabbi  2647  rabxfrd  4452  asymref  4994  rexrnmpt  5637  mpo2eqb  5960
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