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Theorem an6 1264
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 799 . . 3  |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  ( ( th  /\  ta )  /\  et ) )  <->  ( ( (
ph  /\  ps )  /\  ( th  /\  ta ) )  /\  ( ch  /\  et ) ) )
2 an4 799 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( th  /\  ta ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )
) )
32anbi1i 678 . . 3  |-  ( ( ( ( ph  /\  ps )  /\  ( th  /\  ta ) )  /\  ( ch  /\  et ) )  <->  ( (
( ph  /\  th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
41, 3bitri 242 . 2  |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  ( ( th  /\  ta )  /\  et ) )  <->  ( ( (
ph  /\  th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
5 df-3an 939 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
6 df-3an 939 . . 3  |-  ( ( th  /\  ta  /\  et )  <->  ( ( th 
/\  ta )  /\  et ) )
75, 6anbi12i 680 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( (
( ph  /\  ps )  /\  ch )  /\  (
( th  /\  ta )  /\  et ) ) )
8 df-3an 939 . 2  |-  ( ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) )  <-> 
( ( ( ph  /\ 
th )  /\  ( ps  /\  ta ) )  /\  ( ch  /\  et ) ) )
94, 7, 83bitr4i 270 1  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  ( th  /\  ta  /\  et ) )  <->  ( ( ph  /\  th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937
This theorem is referenced by:  3an6  1265  ltdiv2OLD  9927  elfzuzb  11084  ptbasin  17640  iimulcl  18993  nb3grapr  21493  nb3grapr2  21494  txpcon  24950  fzadd2  26483  paddasslem9  30723  paddasslem10  30724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939
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