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Theorem prtlem70 26715
Description: Lemma for prter3 26750: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
Assertion
Ref Expression
prtlem70  |-  ( ( ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ( ph  /\  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) )  /\  et ) )

Proof of Theorem prtlem70
StepHypRef Expression
1 anass 630 . . . 4  |-  ( ( ( ( ph  /\  ps )  /\  ( ph  /\  th ) )  /\  ( ch  /\  ta ) )  <->  ( ( ph  /\  ps )  /\  ( ( ph  /\  th )  /\  ( ch 
/\  ta ) ) ) )
21anbi1i 676 . . 3  |-  ( ( ( ( ( ph  /\ 
ps )  /\  ( ph  /\  th ) )  /\  ( ch  /\  ta ) )  /\  et ) 
<->  ( ( ( ph  /\ 
ps )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  et ) )
3 anandi 801 . . . . 5  |-  ( (
ph  /\  ( ps  /\ 
th ) )  <->  ( ( ph  /\  ps )  /\  ( ph  /\  th )
) )
43anbi1i 676 . . . 4  |-  ( ( ( ph  /\  ( ps  /\  th ) )  /\  ( ch  /\  ta ) )  <->  ( (
( ph  /\  ps )  /\  ( ph  /\  th ) )  /\  ( ch  /\  ta ) ) )
54anbi1i 676 . . 3  |-  ( ( ( ( ph  /\  ( ps  /\  th )
)  /\  ( ch  /\ 
ta ) )  /\  et )  <->  ( ( ( ( ph  /\  ps )  /\  ( ph  /\  th ) )  /\  ( ch  /\  ta ) )  /\  et ) )
6 anass 630 . . . . 5  |-  ( ( ( ph  /\  ( ps  /\  et ) )  /\  ( ( ph  /\ 
th )  /\  ( ch  /\  ta ) ) )  <->  ( ph  /\  ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) ) ) )
7 anass 630 . . . . . 6  |-  ( ( ( ph  /\  ps )  /\  et )  <->  ( ph  /\  ( ps  /\  et ) ) )
87anbi1i 676 . . . . 5  |-  ( ( ( ( ph  /\  ps )  /\  et )  /\  ( ( ph  /\ 
th )  /\  ( ch  /\  ta ) ) )  <->  ( ( ph  /\  ( ps  /\  et ) )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) ) )
9 ancom 437 . . . . 5  |-  ( ( ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ph  /\  ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) ) ) )
106, 8, 93bitr4ri 269 . . . 4  |-  ( ( ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ( (
ph  /\  ps )  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) ) )
11 ancom 437 . . . . 5  |-  ( ( ( ph  /\  ps )  /\  et )  <->  ( et  /\  ( ph  /\  ps ) ) )
1211anbi1i 676 . . . 4  |-  ( ( ( ( ph  /\  ps )  /\  et )  /\  ( ( ph  /\ 
th )  /\  ( ch  /\  ta ) ) )  <->  ( ( et 
/\  ( ph  /\  ps ) )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) ) )
13 anass 630 . . . . 5  |-  ( ( ( et  /\  ( ph  /\  ps ) )  /\  ( ( ph  /\ 
th )  /\  ( ch  /\  ta ) ) )  <->  ( et  /\  ( ( ph  /\  ps )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) ) ) )
14 ancom 437 . . . . 5  |-  ( ( et  /\  ( (
ph  /\  ps )  /\  ( ( ph  /\  th )  /\  ( ch 
/\  ta ) ) ) )  <->  ( ( (
ph  /\  ps )  /\  ( ( ph  /\  th )  /\  ( ch 
/\  ta ) ) )  /\  et ) )
1513, 14bitri 240 . . . 4  |-  ( ( ( et  /\  ( ph  /\  ps ) )  /\  ( ( ph  /\ 
th )  /\  ( ch  /\  ta ) ) )  <->  ( ( (
ph  /\  ps )  /\  ( ( ph  /\  th )  /\  ( ch 
/\  ta ) ) )  /\  et ) )
1610, 12, 153bitri 262 . . 3  |-  ( ( ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ( (
ph  /\  ps )  /\  ( ( ph  /\  th )  /\  ( ch 
/\  ta ) ) )  /\  et ) )
172, 5, 163bitr4ri 269 . 2  |-  ( ( ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ( (
ph  /\  ( ps  /\ 
th ) )  /\  ( ch  /\  ta )
)  /\  et )
)
18 anass 630 . . 3  |-  ( ( ( ph  /\  ( ps  /\  th ) )  /\  ( ch  /\  ta ) )  <->  ( ph  /\  ( ( ps  /\  th )  /\  ( ch 
/\  ta ) ) ) )
1918anbi1i 676 . 2  |-  ( ( ( ( ph  /\  ( ps  /\  th )
)  /\  ( ch  /\ 
ta ) )  /\  et )  <->  ( ( ph  /\  ( ( ps  /\  th )  /\  ( ch 
/\  ta ) ) )  /\  et ) )
20 an4 797 . . . . 5  |-  ( ( ( ps  /\  th )  /\  ( ch  /\  ta ) )  <->  ( ( ps  /\  ch )  /\  ( th  /\  ta )
) )
21 anass 630 . . . . 5  |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  <->  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) )
2220, 21bitri 240 . . . 4  |-  ( ( ( ps  /\  th )  /\  ( ch  /\  ta ) )  <->  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) )
2322anbi2i 675 . . 3  |-  ( (
ph  /\  ( ( ps  /\  th )  /\  ( ch  /\  ta )
) )  <->  ( ph  /\  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) ) )
2423anbi1i 676 . 2  |-  ( ( ( ph  /\  (
( ps  /\  th )  /\  ( ch  /\  ta ) ) )  /\  et )  <->  ( ( ph  /\  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) )  /\  et ) )
2517, 19, 243bitri 262 1  |-  ( ( ( ( ps  /\  et )  /\  (
( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ( ph  /\  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) )  /\  et ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358
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
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