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Theorem rexsns 3671
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 3015 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
21a1i 10 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
3 df-rex 2549 . . 3  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
4 elsn 3655 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
54anbi1i 676 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
65exbii 1569 . . 3  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
73, 6bitri 240 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  =  A  /\  ph ) )
82, 7syl6rbbr 255 1  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   [.wsbc 2991   {csn 3640
This theorem is referenced by:  rexsng  3673  r19.12sn  3696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-sn 3646
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