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Theorem rexsns 4060
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 4041 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21anbi1i 695 . . 3  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
32exbii 1644 . 2  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
4 df-rex 2820 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
5 sbc5 3356 . 2  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
63, 4, 53bitr4i 277 1  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   [.wsbc 3331   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-rex 2820  df-v 3115  df-sbc 3332  df-sn 4028
This theorem is referenced by:  rexsng  4063
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