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Theorem ceqsexv 2836
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
Hypotheses
Ref Expression
ceqsexv.1  |-  A  e. 
_V
ceqsexv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsexv  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsexv
StepHypRef Expression
1 nfv 1609 . 2  |-  F/ x ps
2 ceqsexv.1 . 2  |-  A  e. 
_V
3 ceqsexv.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3ceqsex 2835 1  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  ceqsex3v  2839  gencbvex  2843  sbhypf  2846  euxfr2  2963  inuni  4189  eqvinop  4267  dfid3  4326  uniuni  4543  opeliunxp  4756  elvvv  4765  imai  5043  elxp4  5176  elxp5  5177  coi1  5204  dmfco  5609  fndmdif  5645  fndmin  5648  fmptco  5707  abrexco  5782  opabex3  5785  fsplit  6239  brtpos2  6256  mapsnen  6954  xpsnen  6962  xpcomco  6968  xpassen  6972  dfac5lem1  7766  dfac5lem2  7767  dfac5lem3  7768  cf0  7893  ltexprlem4  8679  pceu  12915  4sqlem12  13019  vdwapun  13037  gsumval3eu  15206  dprd2d2  15295  znleval  16524  metrest  18086  ceqsexv2d  23178  fmptcof2  23244  dfdm5  24203  dfrn5  24204  nofulllem5  24431  brtxp  24491  brpprod  24496  dffun10  24524  dfiota3  24533  brimg  24547  brapply  24548  brcup  24549  brcap  24550  brsuccf  24551  funpartlem  24552  brrestrict  24559  dfrdg4  24560  tfrqfree  24561  eqvinopb  25068  diophrex  26958  opabex3d  28190  lshpsmreu  29921  isopos  29992  islpln5  30346  islvol5  30390  pmapglb  30581  polval2N  30717  cdlemftr3  31376  dibelval3  31959  dicelval3  31992  mapdpglem3  32487  hdmapglem7a  32742
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-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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