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Theorem rexsns 3486
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 3467 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21anbi1i 447 . . 3  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
32exbii 1542 . 2  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
4 df-rex 2366 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
5 sbc5 2864 . 2  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
63, 4, 53bitr4i 211 1  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1290   E.wex 1427    e. wcel 1439   E.wrex 2361   [.wsbc 2841   {csn 3450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-sbc 2842  df-sn 3456
This theorem is referenced by:  rexsng  3488  r19.12sn  3512  iunxsngf  3813  finexdc  6672  exfzdc  9705
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