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Theorem rspce 3049
 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 2574 . . . 4
2 nfv 1630 . . . . 5
3 rspc.1 . . . . 5
42, 3nfan 1847 . . . 4
5 eleq1 2498 . . . . 5
6 rspc.2 . . . . 5
75, 6anbi12d 693 . . . 4
81, 4, 7spcegf 3034 . . 3
98anabsi5 792 . 2
10 df-rex 2713 . 2
119, 10sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551  wnf 1554   wceq 1653   wcel 1726  wrex 2708 This theorem is referenced by:  rspcev  3054  ac6c4  8366  fsumcom2  12563  infcvgaux1i  12641  iunmbl2  19456  esumcvg  24481  fprodcom2  25313  sdclem1  26461 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960
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