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Theorem ralsnsg 3569
 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 df-ral 2422 . . 3
2 velsn 3550 . . . . 5
32imbi1i 237 . . . 4
43albii 1447 . . 3
51, 4bitri 183 . 2
6 sbc6g 2938 . 2
75, 6bitr4id 198 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104  wal 1330   wceq 1332   wcel 1481  wral 2417  wsbc 2914  csn 3533 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2692  df-sbc 2915  df-sn 3539 This theorem is referenced by:  ixpsnval  6604  ac6sfi  6801  rexfiuz  10813  prmind2  11857
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