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Theorem rspcdv 3600
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1 (𝜑𝐴𝐵)
rspcdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rspcdv (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdv
StepHypRef Expression
1 rspcdv.1 . 2 (𝜑𝐴𝐵)
2 rspcdv.2 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
32biimpd 228 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
41, 3rspcimdv 3598 1 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052
This theorem is referenced by:  rspcdv2  3603  rspcv  3604  ralxfrd  5403  ralxfrd2  5407  reuop  6295  suppofss1d  8209  suppofss2d  8210  zindd  12707  wrd2ind  14724  ismri2dad  17643  mreexd  17648  mreexexlemd  17650  catcocl  17691  catass  17692  moni  17745  subccocl  17857  funcco  17883  fullfo  17927  fthf1  17932  nati  17971  mndind  18811  ringurd  20162  idsrngd  20829  mpomulcn  24871  fsumdvdsmul  27218  uspgr2wlkeq  29578  crctcshwlkn0lem4  29742  crctcshwlkn0lem5  29743  wwlknllvtx  29775  0enwwlksnge1  29793  wlkiswwlks2lem5  29802  clwlkclwwlklem2a  29926  clwlkclwwlklem2  29928  clwwisshclwws  29943  clwwlkinwwlk  29968  umgr2cwwk2dif  29992  wrdt2ind  32818  mgccole1  32861  mgccole2  32862  mgcmnt1  32863  mgcmntco  32865  dfmgc2lem  32866  chnind  32881  1arithufdlem3  33425  dfufd2  33429  fedgmullem2  33529  constrconj  33615  zart0  33705  zarcmplem  33707  esumcvg  33930  inelcarsg  34156  carsgclctunlem1  34162  orvcelel  34314  signsply0  34408  onint1  36172  qsalrel  41984  ismnushort  44010  ralbinrald  46769  fargshiftfva  47049  reupr  47128  evengpop3  47404  evengpoap3  47405  snlindsntorlem  47887
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