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Theorem rspce 2811
 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 2299 . . . 4
2 nfv 1508 . . . . 5
3 rspc.1 . . . . 5
42, 3nfan 1545 . . . 4
5 eleq1 2220 . . . . 5
6 rspc.2 . . . . 5
75, 6anbi12d 465 . . . 4
81, 4, 7spcegf 2795 . . 3
98anabsi5 569 . 2
10 df-rex 2441 . 2
119, 10sylibr 133 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1335  wnf 1440  wex 1472   wcel 2128  wrex 2436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714 This theorem is referenced by:  rspcev  2816  bezoutlemmain  11873
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