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Theorem ralsns 3408
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsns
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem ralsns
StepHypRef Expression
1 sbc6g 2788 . 2
2 df-ral 2311 . . 3
3 velsn 3392 . . . . 5
43imbi1i 227 . . . 4
54albii 1359 . . 3
62, 5bitri 173 . 2
71, 6syl6rbbr 188 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98  wal 1241   wceq 1243   wcel 1393  wral 2306  wsbc 2764  csn 3375 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-ral 2311  df-v 2559  df-sbc 2765  df-sn 3381 This theorem is referenced by:  ralsng  3411  sbcsng  3429  rabrsndc  3438
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