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Theorem rspccva 2855
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
rspcv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rspccva  |-  ( ( A. x  e.  B  ph 
/\  A  e.  B
)  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rspccva
StepHypRef Expression
1 rspcv.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21rspcv 2852 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) )
32impcom 125 1  |-  ( ( A. x  e.  B  ph 
/\  A  e.  B
)  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754
This theorem is referenced by:  disjne  3491  seex  4353  fconstfvm  5755  fvixp  6729  ordiso2  7064  eqord1  8470  eqord2  8471  seq3caopr2  10513  bccl  10779  2clim  11341  isummulc2  11466  telfsumo2  11507  fsumparts  11510  isumshft  11530  mertenslem2  11576  mertensabs  11577  dvdsprime  12154  mgmlrid  12855  grpinvalem  12861  grpinvex  12955  issubg2m  13128  issubg4m  13132  nmzbi  13148  cnima  14177  dceqnconst  15267  dcapnconst  15268
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