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Theorem elabgt 2825
 Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2830.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
elabgt
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem elabgt
StepHypRef Expression
1 abid 2127 . . . . . . 7
2 eleq1 2202 . . . . . . 7
31, 2bitr3id 193 . . . . . 6
43bibi1d 232 . . . . 5
54biimpd 143 . . . 4
65a2i 11 . . 3
76alimi 1431 . 2
8 nfcv 2281 . . . 4
9 nfab1 2283 . . . . . 6
109nfel2 2294 . . . . 5
11 nfv 1508 . . . . 5
1210, 11nfbi 1568 . . . 4
13 pm5.5 241 . . . 4
148, 12, 13spcgf 2768 . . 3
1514imp 123 . 2
167, 15sylan2 284 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1329   wceq 1331   wcel 1480  cab 2125 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688 This theorem is referenced by:  elrab3t  2839
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