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Theorem fatesg 24956
Description: Equivalence of  A. and  E. in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
Assertion
Ref Expression
fatesg  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  E. x  e.  { A } ph ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem fatesg
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 sneq 3651 . . 3  |-  ( a  =  A  ->  { a }  =  { A } )
21raleqdv 2742 . 2  |-  ( a  =  A  ->  ( A. x  e.  { a } ph  <->  A. x  e.  { A } ph ) )
31rexeqdv 2743 . 2  |-  ( a  =  A  ->  ( E. x  e.  { a } ph  <->  E. x  e.  { A } ph ) )
4 vex 2791 . . 3  |-  a  e. 
_V
54fates 24955 . 2  |-  ( A. x  e.  { a } ph  <->  E. x  e.  {
a } ph )
62, 3, 5vtoclbg 2844 1  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  E. x  e.  { A } ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   {csn 3640
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-ral 2548  df-rex 2549  df-v 2790  df-sn 3646
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