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Theorem ralsns 3399
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsns  V  { }  [.  ].
Distinct variable group:   ,
Allowed substitution hints:   ()    V()

Proof of Theorem ralsns
StepHypRef Expression
1 sbc6g 2782 . 2  V  [.  ].
2 df-ral 2305 . . 3  { }  { }
3 elsn 3382 . . . . 5  { }
43imbi1i 227 . . . 4  { }
54albii 1356 . . 3  { }
62, 5bitri 173 . 2  { }
71, 6syl6rbbr 188 1  V  { }  [.  ].
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98  wal 1240   wceq 1242   wcel 1390  wral 2300   [.wsbc 2758   {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-sn 3373
This theorem is referenced by:  ralsng  3402  sbcsng  3420  rabrsndc  3429
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