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Theorem rspce 3214
 Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
rspc.1
rspc.2
Assertion
Ref Expression
rspce
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem rspce
StepHypRef Expression
1 nfcv 2629 . . . 4
2 nfv 1683 . . . . 5
3 rspc.1 . . . . 5
42, 3nfan 1875 . . . 4
5 eleq1 2539 . . . . 5
6 rspc.2 . . . . 5
75, 6anbi12d 710 . . . 4
81, 4, 7spcegf 3199 . . 3
98anabsi5 815 . 2
10 df-rex 2823 . 2
119, 10sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379  wex 1596  wnf 1599   wcel 1767  wrex 2818 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-v 3120 This theorem is referenced by:  rspcev  3219  ac6c4  8873  fsumcom2  13569  infcvgaux1i  13648  iunmbl2  21835  esumcvg  27917  fprodcom2  29041  sdclem1  30163
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