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Theorem rexsns 4035
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 elsn 4016 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21anbi1i 699 . . 3  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
32exbii 1714 . 2  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
4 df-rex 2788 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
5 sbc5 3330 . 2  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
63, 4, 53bitr4i 280 1  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   E.wrex 2783   [.wsbc 3305   {csn 4002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-rex 2788  df-v 3089  df-sbc 3306  df-sn 4003
This theorem is referenced by:  rexsng  4038  r19.12sn  4068  poimirlem25  31669  rexsngf  37032
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