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Theorem ralsnsg 3438
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsnsg
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem ralsnsg
StepHypRef Expression
1 sbc6g 2840 . 2
2 df-ral 2354 . . 3
3 velsn 3423 . . . . 5
43imbi1i 236 . . . 4
54albii 1400 . . 3
62, 5bitri 182 . 2
71, 6syl6rbbr 197 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 103  wal 1283   wceq 1285   wcel 1434  wral 2349  wsbc 2816  csn 3406 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-sbc 2817  df-sn 3412 This theorem is referenced by:  ac6sfi  6431  rexfiuz  10013  prmind2  10646
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