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Theorem sbc5 2787
 Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
sbc5
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem sbc5
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcex 2772 . 2
2 exsimpl 1508 . . 3
3 isset 2561 . . 3
42, 3sylibr 137 . 2
5 dfsbcq2 2767 . . 3
6 eqeq2 2049 . . . . 5
76anbi1d 438 . . . 4
87exbidv 1706 . . 3
9 sb5 1767 . . 3
105, 8, 9vtoclbg 2614 . 2
111, 4, 10pm5.21nii 620 1
 Colors of variables: wff set class Syntax hints:   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393  wsb 1645  cvv 2557  wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765 This theorem is referenced by:  sbc6g  2788  sbc7  2790  sbciegft  2793  sbccomlem  2832  csb2  2854  rexsns  3409  rexsnsOLD  3410
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