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Theorem rspcdv 3081
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1  |-  ( ph  ->  A  e.  B )
rspcdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rspcdv  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcdv
StepHypRef Expression
1 rspcdv.1 . 2  |-  ( ph  ->  A  e.  B )
2 rspcdv.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32biimpd 207 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
41, 3rspcimdv 3079 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-v 2979
This theorem is referenced by:  ralxfrd  4511  suppofss1d  6731  suppofss2d  6732  zindd  10748  wrd2ind  12377  ismri2dad  14580  mreexd  14585  mreexexlemd  14587  catcocl  14628  catass  14629  moni  14680  subccocl  14760  funcco  14786  fullfo  14827  fthf1  14832  nati  14870  mrcmndind  15499  idsrngd  16952  sizeusglecusglem1  23397  fargshiftfva  23530  eupatrl  23594  rngurd  26261  esumcvg  26540  orvcelel  26857  signsply0  26957  onint1  28300  ralbinrald  30028  ralxfrd2  30142  usg2wlkeq  30294  wlkiswwlk2lem5  30334  clwlkisclwwlklem2a  30452  clwlkisclwwlklem1  30454  clwwisshclww  30476  usg2cwwk2dif  30499
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