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Theorem rexsns 3805
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 3145 . . 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 2672 . . 3  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
4 elsn 3789 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
54anbi1i 677 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
65exbii 1589 . . 3  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
73, 6bitri 241 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  =  A  /\  ph ) )
82, 7syl6rbbr 256 1  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2667   [.wsbc 3121   {csn 3774
This theorem is referenced by:  rexsng  3807  r19.12sn  3832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-v 2918  df-sbc 3122  df-sn 3780
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