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Theorem rexsns 3620
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 3598 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21anbi1i 455 . . 3  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
32exbii 1598 . 2  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
4 df-rex 2454 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
5 sbc5 2978 . 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 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   [.wsbc 2955   {csn 3581
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-sn 3587
This theorem is referenced by:  rexsng  3622  r19.12sn  3647  iunxsngf  3948  finexdc  6878  exfzdc  10189
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