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Theorem keephyp3v 3634
Description: Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
Hypotheses
Ref Expression
keephyp3v.1  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( rh  <->  ch )
)
keephyp3v.2  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
keephyp3v.3  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
keephyp3v.4  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
keephyp3v.5  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
keephyp3v.6  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta )
)
keephyp3v.7  |-  rh
keephyp3v.8  |-  et
Assertion
Ref Expression
keephyp3v  |-  ta

Proof of Theorem keephyp3v
StepHypRef Expression
1 keephyp3v.7 . . 3  |-  rh
2 iftrue 3584 . . . . . 6  |-  ( ph  ->  if ( ph ,  A ,  D )  =  A )
32eqcomd 2301 . . . . 5  |-  ( ph  ->  A  =  if (
ph ,  A ,  D ) )
4 keephyp3v.1 . . . . 5  |-  ( A  =  if ( ph ,  A ,  D )  ->  ( rh  <->  ch )
)
53, 4syl 15 . . . 4  |-  ( ph  ->  ( rh  <->  ch )
)
6 iftrue 3584 . . . . . 6  |-  ( ph  ->  if ( ph ,  B ,  R )  =  B )
76eqcomd 2301 . . . . 5  |-  ( ph  ->  B  =  if (
ph ,  B ,  R ) )
8 keephyp3v.2 . . . . 5  |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch  <->  th )
)
97, 8syl 15 . . . 4  |-  ( ph  ->  ( ch  <->  th )
)
10 iftrue 3584 . . . . . 6  |-  ( ph  ->  if ( ph ,  C ,  S )  =  C )
1110eqcomd 2301 . . . . 5  |-  ( ph  ->  C  =  if (
ph ,  C ,  S ) )
12 keephyp3v.3 . . . . 5  |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta )
)
1311, 12syl 15 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
145, 9, 133bitrd 270 . . 3  |-  ( ph  ->  ( rh  <->  ta )
)
151, 14mpbii 202 . 2  |-  ( ph  ->  ta )
16 keephyp3v.8 . . 3  |-  et
17 iffalse 3585 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  A ,  D )  =  D )
1817eqcomd 2301 . . . . 5  |-  ( -. 
ph  ->  D  =  if ( ph ,  A ,  D ) )
19 keephyp3v.4 . . . . 5  |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze )
)
2018, 19syl 15 . . . 4  |-  ( -. 
ph  ->  ( et  <->  ze )
)
21 iffalse 3585 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  B ,  R )  =  R )
2221eqcomd 2301 . . . . 5  |-  ( -. 
ph  ->  R  =  if ( ph ,  B ,  R ) )
23 keephyp3v.5 . . . . 5  |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si )
)
2422, 23syl 15 . . . 4  |-  ( -. 
ph  ->  ( ze  <->  si )
)
25 iffalse 3585 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  C ,  S )  =  S )
2625eqcomd 2301 . . . . 5  |-  ( -. 
ph  ->  S  =  if ( ph ,  C ,  S ) )
27 keephyp3v.6 . . . . 5  |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta )
)
2826, 27syl 15 . . . 4  |-  ( -. 
ph  ->  ( si  <->  ta )
)
2920, 24, 283bitrd 270 . . 3  |-  ( -. 
ph  ->  ( et  <->  ta )
)
3016, 29mpbii 202 . 2  |-  ( -. 
ph  ->  ta )
3115, 30pm2.61i 156 1  |-  ta
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1632   ifcif 3578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
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