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Theorem r19.41v 2706
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
Assertion
Ref Expression
r19.41v  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ph 
/\  ps ) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem r19.41v
StepHypRef Expression
1 nfv 1609 . 2  |-  F/ x ps
21r19.41 2705 1  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ph 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wrex 2557
This theorem is referenced by:  r19.42v  2707  3reeanv  2721  reuind  2981  iuncom4  3928  dfiun2g  3951  iunxiun  4000  inuni  4189  reuxfrd  4575  xpiundi  4759  xpiundir  4760  imaco  5194  coiun  5198  abrexco  5782  imaiun  5787  isomin  5850  isoini  5851  oarec  6576  mapsnen  6954  genpass  8649  4sqlem12  13019  imasleval  13459  lsmspsn  15853  metrest  18086  nmoo0  21385  hhcmpl  21795  nmop0  22582  nmfn0  22583  reuxfr4d  23155  rexunirn  23156  r19.41vv  23179  nofulllem5  24431  elfuns  24525  axsegcon  24627  axeuclid  24663  rabiun2  24997  itg2addnc  25005  prtlem11  26837  prter2  26852  prter3  26853  diophrex  26958  4fvwrd4  28220  islshpat  29829  lshpsmreu  29921  islpln5  30346  islvol5  30390  cdlemftr3  31376  dvhb1dimN  31797  dib1dim  31977  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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-rex 2562
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