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Theorem elimhyp 3943
Description: Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3936. (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 3891 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21eqcomd 2410 . . . 4  |-  ( ph  ->  A  =  if (
ph ,  A ,  B ) )
3 elimhyp.1 . . . 4  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps )
)
42, 3syl 17 . . 3  |-  ( ph  ->  ( ph  <->  ps )
)
54ibi 241 . 2  |-  ( ph  ->  ps )
6 elimhyp.3 . . 3  |-  ch
7 iffalse 3894 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
87eqcomd 2410 . . . 4  |-  ( -. 
ph  ->  B  =  if ( ph ,  A ,  B ) )
9 elimhyp.2 . . . 4  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  ps )
)
108, 9syl 17 . . 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 1405   ifcif 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-if 3886
This theorem is referenced by:  elimel  3947  elimf  5713  oeoa  7283  oeoe  7285  limensuc  7732  axcc4dom  8853  elimne0  9616  elimgt0  10419  elimge0  10420  2ndcdisj  20249  siilem2  26181  normlem7tALT  26450  hhsssh  26599  shintcl  26662  chintcl  26664  spanun  26877  elunop2  27345  lnophm  27351  nmbdfnlb  27382  hmopidmch  27485  hmopidmpj  27486  chirred  27727  limsucncmp  30678  elimhyps  31985  elimhyps2  31988
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