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Theorem an6 1261
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 an4 797 . . 3  |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  ( ( th  /\  ta )  /\  et ) )  <->  ( ( (
ph  /\  ps )  /\  ( th  /\  ta ) )  /\  ( ch  /\  et ) ) )
2 an4 797 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( th  /\  ta ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )
) )
32anbi1i 676 . . 3  |-  ( ( ( ( ph  /\  ps )  /\  ( th  /\  ta ) )  /\  ( ch  /\  et ) )  <->  ( (
( ph  /\  th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
41, 3bitri 240 . 2  |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  ( ( th  /\  ta )  /\  et ) )  <->  ( ( (
ph  /\  th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
5 df-3an 936 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
6 df-3an 936 . . 3  |-  ( ( th  /\  ta  /\  et )  <->  ( ( th 
/\  ta )  /\  et ) )
75, 6anbi12i 678 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( (
( ph  /\  ps )  /\  ch )  /\  (
( th  /\  ta )  /\  et ) ) )
8 df-3an 936 . 2  |-  ( ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) )  <-> 
( ( ( ph  /\ 
th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
94, 7, 83bitr4i 268 1  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934
This theorem is referenced by:  3an6  1262  ltdiv2OLD  9644  elfzuzb  10794  ptbasin  17274  iimulcl  18437  txpcon  23765  limptlimpr2lem2  25586  fzadd2  26455  paddasslem9  30090  paddasslem10  30091
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
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