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Theorem keepel 3635
Description: Keep a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 14-Aug-1999.)
Hypotheses
Ref Expression
keepel.1  |-  A  e.  C
keepel.2  |-  B  e.  C
Assertion
Ref Expression
keepel  |-  if (
ph ,  A ,  B )  e.  C

Proof of Theorem keepel
StepHypRef Expression
1 eleq1 2356 . 2  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( A  e.  C  <->  if ( ph ,  A ,  B )  e.  C ) )
2 eleq1 2356 . 2  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( B  e.  C  <->  if ( ph ,  A ,  B )  e.  C ) )
3 keepel.1 . 2  |-  A  e.  C
4 keepel.2 . 2  |-  B  e.  C
51, 2, 3, 4keephyp 3632 1  |-  if (
ph ,  A ,  B )  e.  C
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   ifcif 3578
This theorem is referenced by:  ifex  3636  xaddf  10567  ccatfn  11443  sadcf  12660  ramcl  13092  setcepi  13936  abvtrivd  15621  mvridlem  16180  mvrf1  16186  mplcoe3  16226  psrbagsn  16252  dscmet  18111  dscopn  18112  i1f1lem  19060  i1f1  19061  itg2const  19111  evlslem1  19415  cxpval  20027  cxpcl  20037  recxpcl  20038  sqff1o  20436  chtublem  20466  dchrmulid2  20507  bposlem1  20539  lgsval  20555  lgsfcl2  20557  lgscllem  20558  lgsval2lem  20561  lgsneg  20574  lgsdilem  20577  lgsdir2  20583  lgsdir  20585  lgsdi  20587  lgsne0  20588  dchrisum0flblem1  20673  dchrisum0flblem2  20674  dchrisum0fno1  20676  rpvmasum2  20677  omlsi  21999  indfval  23615  sqdivzi  24094  pw2f1ocnv  27233  flcidc  27482
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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