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Theorem an6 1227
Description: Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
Assertion
Ref Expression
an6  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) ) )

Proof of Theorem an6
StepHypRef Expression
1 df-3an 898 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
2 df-3an 898 . . . 4  |-  ( ( th  /\  ta  /\  et )  <->  ( ( th 
/\  ta )  /\  et ) )
31, 2anbi12i 441 . . 3  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( (
( ph  /\  ps )  /\  ch )  /\  (
( th  /\  ta )  /\  et ) ) )
4 an4 528 . . 3  |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  ( ( th  /\  ta )  /\  et ) )  <->  ( ( (
ph  /\  ps )  /\  ( th  /\  ta ) )  /\  ( ch  /\  et ) ) )
5 an4 528 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( th  /\  ta ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )
) )
65anbi1i 439 . . 3  |-  ( ( ( ( ph  /\  ps )  /\  ( th  /\  ta ) )  /\  ( ch  /\  et ) )  <->  ( (
( ph  /\  th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
73, 4, 63bitri 199 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( (
( ph  /\  th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
8 df-3an 898 . 2  |-  ( ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) )  <-> 
( ( ( ph  /\ 
th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
97, 8bitr4i 180 1  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    /\ w3a 896
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  df-3an 898
This theorem is referenced by:  3an6  1228  elfzuzb  8986
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