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Theorem elimhyp 3732
Description: Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3725. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
elimhyp.1  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps )
)
elimhyp.2  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  ps )
)
elimhyp.3  |-  ch
Assertion
Ref Expression
elimhyp  |-  ps

Proof of Theorem elimhyp
StepHypRef Expression
1 iftrue 3690 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21eqcomd 2394 . . . 4  |-  ( ph  ->  A  =  if (
ph ,  A ,  B ) )
3 elimhyp.1 . . . 4  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps )
)
42, 3syl 16 . . 3  |-  ( ph  ->  ( ph  <->  ps )
)
54ibi 233 . 2  |-  ( ph  ->  ps )
6 elimhyp.3 . . 3  |-  ch
7 iffalse 3691 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
87eqcomd 2394 . . . 4  |-  ( -. 
ph  ->  B  =  if ( ph ,  A ,  B ) )
9 elimhyp.2 . . . 4  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  ps )
)
108, 9syl 16 . . 3  |-  ( -. 
ph  ->  ( ch  <->  ps )
)
116, 10mpbii 203 . 2  |-  ( -. 
ph  ->  ps )
125, 11pm2.61i 158 1  |-  ps
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649   ifcif 3684
This theorem is referenced by:  elimel  3736  elimf  5532  oeoa  6778  oeoe  6780  limensuc  7222  axcc4dom  8256  elimne0  9017  elimgt0  9780  elimge0  9781  2ndcdisj  17442  siilem2  22203  normlem7tALT  22471  hhsssh  22619  shintcl  22682  chintcl  22684  spanun  22897  elunop2  23366  lnophm  23372  nmbdfnlb  23403  hmopidmch  23506  hmopidmpj  23507  chirred  23748  limsucncmp  25912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-if 3685
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