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Theorem 3impexpbicom 1357
Description: 3impexp 1356 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD 28948. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
Assertion
Ref Expression
3impexpbicom  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  ( th 
<->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th )
) ) ) )

Proof of Theorem 3impexpbicom
StepHypRef Expression
1 bicom 191 . . . 4  |-  ( ( th  <->  ta )  <->  ( ta  <->  th ) )
2 imbi2 314 . . . . 5  |-  ( ( ( th  <->  ta )  <->  ( ta  <->  th ) )  -> 
( ( ( ph  /\ 
ps  /\  ch )  ->  ( th  <->  ta )
)  <->  ( ( ph  /\ 
ps  /\  ch )  ->  ( ta  <->  th )
) ) )
32biimpcd 215 . . . 4  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  ( th 
<->  ta ) )  -> 
( ( ( th  <->  ta )  <->  ( ta  <->  th )
)  ->  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th )
) ) )
41, 3mpi 16 . . 3  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  ( th 
<->  ta ) )  -> 
( ( ph  /\  ps  /\  ch )  -> 
( ta  <->  th )
) )
543expd 1168 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  ( th 
<->  ta ) )  -> 
( ph  ->  ( ps 
->  ( ch  ->  ( ta 
<->  th ) ) ) ) )
6 3impexp 1356 . . . 4  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  ( ta 
<->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th )
) ) ) )
76biimpri 197 . . 3  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->  ( ( ph  /\ 
ps  /\  ch )  ->  ( ta  <->  th )
) )
87, 1syl6ibr 218 . 2  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->  ( ( ph  /\ 
ps  /\  ch )  ->  ( th  <->  ta )
) )
95, 8impbii 180 1  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  ( th 
<->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934
This theorem is referenced by:  3impexpbicomiVD  28950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
  Copyright terms: Public domain W3C validator