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Theorem elimhyp4v 4006
Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3996). (Contributed by NM, 16-Apr-2005.)
Hypotheses
Ref Expression
elimhyp4v.1  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch )
)
elimhyp4v.2  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
elimhyp4v.3  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
elimhyp4v.4  |-  ( F  =  if ( ph ,  F ,  G )  ->  ( ta  <->  ps )
)
elimhyp4v.5  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
elimhyp4v.6  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
elimhyp4v.7  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  rh )
)
elimhyp4v.8  |-  ( G  =  if ( ph ,  F ,  G )  ->  ( rh  <->  ps )
)
elimhyp4v.9  |-  et
Assertion
Ref Expression
elimhyp4v  |-  ps

Proof of Theorem elimhyp4v
StepHypRef Expression
1 iftrue 3950 . . . . . . 7  |-  ( ph  ->  if ( ph ,  A ,  D )  =  A )
21eqcomd 2465 . . . . . 6  |-  ( ph  ->  A  =  if (
ph ,  A ,  D ) )
3 elimhyp4v.1 . . . . . 6  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch )
)
42, 3syl 16 . . . . 5  |-  ( ph  ->  ( ph  <->  ch )
)
5 iftrue 3950 . . . . . . 7  |-  ( ph  ->  if ( ph ,  B ,  R )  =  B )
65eqcomd 2465 . . . . . 6  |-  ( ph  ->  B  =  if (
ph ,  B ,  R ) )
7 elimhyp4v.2 . . . . . 6  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
86, 7syl 16 . . . . 5  |-  ( ph  ->  ( ch  <->  th )
)
94, 8bitrd 253 . . . 4  |-  ( ph  ->  ( ph  <->  th )
)
10 iftrue 3950 . . . . . 6  |-  ( ph  ->  if ( ph ,  C ,  S )  =  C )
1110eqcomd 2465 . . . . 5  |-  ( ph  ->  C  =  if (
ph ,  C ,  S ) )
12 elimhyp4v.3 . . . . 5  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
1311, 12syl 16 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
14 iftrue 3950 . . . . . 6  |-  ( ph  ->  if ( ph ,  F ,  G )  =  F )
1514eqcomd 2465 . . . . 5  |-  ( ph  ->  F  =  if (
ph ,  F ,  G ) )
16 elimhyp4v.4 . . . . 5  |-  ( F  =  if ( ph ,  F ,  G )  ->  ( ta  <->  ps )
)
1715, 16syl 16 . . . 4  |-  ( ph  ->  ( ta  <->  ps )
)
189, 13, 173bitrd 279 . . 3  |-  ( ph  ->  ( ph  <->  ps )
)
1918ibi 241 . 2  |-  ( ph  ->  ps )
20 elimhyp4v.9 . . 3  |-  et
21 iffalse 3953 . . . . . . 7  |-  ( -. 
ph  ->  if ( ph ,  A ,  D )  =  D )
2221eqcomd 2465 . . . . . 6  |-  ( -. 
ph  ->  D  =  if ( ph ,  A ,  D ) )
23 elimhyp4v.5 . . . . . 6  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
2422, 23syl 16 . . . . 5  |-  ( -. 
ph  ->  ( et  <->  ze )
)
25 iffalse 3953 . . . . . . 7  |-  ( -. 
ph  ->  if ( ph ,  B ,  R )  =  R )
2625eqcomd 2465 . . . . . 6  |-  ( -. 
ph  ->  R  =  if ( ph ,  B ,  R ) )
27 elimhyp4v.6 . . . . . 6  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
2826, 27syl 16 . . . . 5  |-  ( -. 
ph  ->  ( ze  <->  si )
)
2924, 28bitrd 253 . . . 4  |-  ( -. 
ph  ->  ( et  <->  si )
)
30 iffalse 3953 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  C ,  S )  =  S )
3130eqcomd 2465 . . . . 5  |-  ( -. 
ph  ->  S  =  if ( ph ,  C ,  S ) )
32 elimhyp4v.7 . . . . 5  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  rh )
)
3331, 32syl 16 . . . 4  |-  ( -. 
ph  ->  ( si  <->  rh )
)
34 iffalse 3953 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  F ,  G )  =  G )
3534eqcomd 2465 . . . . 5  |-  ( -. 
ph  ->  G  =  if ( ph ,  F ,  G ) )
36 elimhyp4v.8 . . . . 5  |-  ( G  =  if ( ph ,  F ,  G )  ->  ( rh  <->  ps )
)
3735, 36syl 16 . . . 4  |-  ( -. 
ph  ->  ( rh  <->  ps )
)
3829, 33, 373bitrd 279 . . 3  |-  ( -. 
ph  ->  ( et  <->  ps )
)
3920, 38mpbii 211 . 2  |-  ( -. 
ph  ->  ps )
4019, 39pm2.61i 164 1  |-  ps
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1395   ifcif 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-if 3945
This theorem is referenced by: (None)
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