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Theorem rexsns 3847
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
rexsns  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem rexsns
StepHypRef Expression
1 sbc5 3187 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
21a1i 11 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
3 df-rex 2713 . . 3  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
4 elsn 3831 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
54anbi1i 678 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
65exbii 1593 . . 3  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
73, 6bitri 242 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  =  A  /\  ph ) )
82, 7syl6rbbr 257 1  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   [.wsbc 3163   {csn 3816
This theorem is referenced by:  rexsng  3849  r19.12sn  3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-sbc 3164  df-sn 3822
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