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Theorem rexsns 3646
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
rexsns  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rexsns
StepHypRef Expression
1 velsn 3624 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21anbi1i 458 . . 3  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
32exbii 1616 . 2  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
4 df-rex 2474 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
5 sbc5 3001 . 2  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
63, 4, 53bitr4i 212 1  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   E.wrex 2469   [.wsbc 2977   {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-sn 3613
This theorem is referenced by:  rexsng  3648  r19.12sn  3673  iunxsngf  3979  finexdc  6931  exfzdc  10272
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