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Theorem elimhyp4v 3616
Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3606). (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 3571 . . . . . . 7  |-  ( ph  ->  if ( ph ,  A ,  D )  =  A )
21eqcomd 2288 . . . . . 6  |-  ( ph  ->  A  =  if (
ph ,  A ,  D ) )
3 elimhyp4v.1 . . . . . 6  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch )
)
42, 3syl 15 . . . . 5  |-  ( ph  ->  ( ph  <->  ch )
)
5 iftrue 3571 . . . . . . 7  |-  ( ph  ->  if ( ph ,  B ,  R )  =  B )
65eqcomd 2288 . . . . . 6  |-  ( ph  ->  B  =  if (
ph ,  B ,  R ) )
7 elimhyp4v.2 . . . . . 6  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
86, 7syl 15 . . . . 5  |-  ( ph  ->  ( ch  <->  th )
)
94, 8bitrd 244 . . . 4  |-  ( ph  ->  ( ph  <->  th )
)
10 iftrue 3571 . . . . . 6  |-  ( ph  ->  if ( ph ,  C ,  S )  =  C )
1110eqcomd 2288 . . . . 5  |-  ( ph  ->  C  =  if (
ph ,  C ,  S ) )
12 elimhyp4v.3 . . . . 5  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
1311, 12syl 15 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
14 iftrue 3571 . . . . . 6  |-  ( ph  ->  if ( ph ,  F ,  G )  =  F )
1514eqcomd 2288 . . . . 5  |-  ( ph  ->  F  =  if (
ph ,  F ,  G ) )
16 elimhyp4v.4 . . . . 5  |-  ( F  =  if ( ph ,  F ,  G )  ->  ( ta  <->  ps )
)
1715, 16syl 15 . . . 4  |-  ( ph  ->  ( ta  <->  ps )
)
189, 13, 173bitrd 270 . . 3  |-  ( ph  ->  ( ph  <->  ps )
)
1918ibi 232 . 2  |-  ( ph  ->  ps )
20 elimhyp4v.9 . . 3  |-  et
21 iffalse 3572 . . . . . . 7  |-  ( -. 
ph  ->  if ( ph ,  A ,  D )  =  D )
2221eqcomd 2288 . . . . . 6  |-  ( -. 
ph  ->  D  =  if ( ph ,  A ,  D ) )
23 elimhyp4v.5 . . . . . 6  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
2422, 23syl 15 . . . . 5  |-  ( -. 
ph  ->  ( et  <->  ze )
)
25 iffalse 3572 . . . . . . 7  |-  ( -. 
ph  ->  if ( ph ,  B ,  R )  =  R )
2625eqcomd 2288 . . . . . 6  |-  ( -. 
ph  ->  R  =  if ( ph ,  B ,  R ) )
27 elimhyp4v.6 . . . . . 6  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
2826, 27syl 15 . . . . 5  |-  ( -. 
ph  ->  ( ze  <->  si )
)
2924, 28bitrd 244 . . . 4  |-  ( -. 
ph  ->  ( et  <->  si )
)
30 iffalse 3572 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  C ,  S )  =  S )
3130eqcomd 2288 . . . . 5  |-  ( -. 
ph  ->  S  =  if ( ph ,  C ,  S ) )
32 elimhyp4v.7 . . . . 5  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  rh )
)
3331, 32syl 15 . . . 4  |-  ( -. 
ph  ->  ( si  <->  rh )
)
34 iffalse 3572 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  F ,  G )  =  G )
3534eqcomd 2288 . . . . 5  |-  ( -. 
ph  ->  G  =  if ( ph ,  F ,  G ) )
36 elimhyp4v.8 . . . . 5  |-  ( G  =  if ( ph ,  F ,  G )  ->  ( rh  <->  ps )
)
3735, 36syl 15 . . . 4  |-  ( -. 
ph  ->  ( rh  <->  ps )
)
3829, 33, 373bitrd 270 . . 3  |-  ( -. 
ph  ->  ( et  <->  ps )
)
3920, 38mpbii 202 . 2  |-  ( -. 
ph  ->  ps )
4019, 39pm2.61i 156 1  |-  ps
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623   ifcif 3565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-if 3566
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