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Theorem elimhyp 3846
Description: Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3839. (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 3795 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21eqcomd 2446 . . . 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 241 . 2  |-  ( ph  ->  ps )
6 elimhyp.3 . . 3  |-  ch
7 iffalse 3797 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
87eqcomd 2446 . . . 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 211 . 2  |-  ( -. 
ph  ->  ps )
125, 11pm2.61i 164 1  |-  ps
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1369   ifcif 3789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-if 3790
This theorem is referenced by:  elimel  3850  elimf  5556  oeoa  7034  oeoe  7036  limensuc  7486  axcc4dom  8608  elimne0  9374  elimgt0  10163  elimge0  10164  2ndcdisj  19058  siilem2  24250  normlem7tALT  24519  hhsssh  24668  shintcl  24731  chintcl  24733  spanun  24946  elunop2  25415  lnophm  25421  nmbdfnlb  25452  hmopidmch  25555  hmopidmpj  25556  chirred  25797  limsucncmp  28290  elimhyps  32609  elimhyps2  32612
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