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Theorem elimhyp3v 4000
Description: Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
Hypotheses
Ref Expression
elimhyp3v.1  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch )
)
elimhyp3v.2  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
elimhyp3v.3  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
elimhyp3v.4  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
elimhyp3v.5  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
elimhyp3v.6  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta )
)
elimhyp3v.7  |-  et
Assertion
Ref Expression
elimhyp3v  |-  ta

Proof of Theorem elimhyp3v
StepHypRef Expression
1 iftrue 3945 . . . . . 6  |-  ( ph  ->  if ( ph ,  A ,  D )  =  A )
21eqcomd 2475 . . . . 5  |-  ( ph  ->  A  =  if (
ph ,  A ,  D ) )
3 elimhyp3v.1 . . . . 5  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch )
)
42, 3syl 16 . . . 4  |-  ( ph  ->  ( ph  <->  ch )
)
5 iftrue 3945 . . . . . 6  |-  ( ph  ->  if ( ph ,  B ,  R )  =  B )
65eqcomd 2475 . . . . 5  |-  ( ph  ->  B  =  if (
ph ,  B ,  R ) )
7 elimhyp3v.2 . . . . 5  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
86, 7syl 16 . . . 4  |-  ( ph  ->  ( ch  <->  th )
)
9 iftrue 3945 . . . . . 6  |-  ( ph  ->  if ( ph ,  C ,  S )  =  C )
109eqcomd 2475 . . . . 5  |-  ( ph  ->  C  =  if (
ph ,  C ,  S ) )
11 elimhyp3v.3 . . . . 5  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
1210, 11syl 16 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
134, 8, 123bitrd 279 . . 3  |-  ( ph  ->  ( ph  <->  ta )
)
1413ibi 241 . 2  |-  ( ph  ->  ta )
15 elimhyp3v.7 . . 3  |-  et
16 iffalse 3948 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  A ,  D )  =  D )
1716eqcomd 2475 . . . . 5  |-  ( -. 
ph  ->  D  =  if ( ph ,  A ,  D ) )
18 elimhyp3v.4 . . . . 5  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
1917, 18syl 16 . . . 4  |-  ( -. 
ph  ->  ( et  <->  ze )
)
20 iffalse 3948 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  B ,  R )  =  R )
2120eqcomd 2475 . . . . 5  |-  ( -. 
ph  ->  R  =  if ( ph ,  B ,  R ) )
22 elimhyp3v.5 . . . . 5  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
2321, 22syl 16 . . . 4  |-  ( -. 
ph  ->  ( ze  <->  si )
)
24 iffalse 3948 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  C ,  S )  =  S )
2524eqcomd 2475 . . . . 5  |-  ( -. 
ph  ->  S  =  if ( ph ,  C ,  S ) )
26 elimhyp3v.6 . . . . 5  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta )
)
2725, 26syl 16 . . . 4  |-  ( -. 
ph  ->  ( si  <->  ta )
)
2819, 23, 273bitrd 279 . . 3  |-  ( -. 
ph  ->  ( et  <->  ta )
)
2915, 28mpbii 211 . 2  |-  ( -. 
ph  ->  ta )
3014, 29pm2.61i 164 1  |-  ta
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1379   ifcif 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-if 3940
This theorem is referenced by:  sseliALT  4578
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