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Theorem fates 24955
Description: Equivalence of  A. and  E. in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
Hypothesis
Ref Expression
fates.1  |-  A  e.  B
Assertion
Ref Expression
fates  |-  ( A. x  e.  { A } ph  <->  E. x  e.  { A } ph )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem fates
StepHypRef Expression
1 fates.1 . . . . 5  |-  A  e.  B
21elexi 2797 . . . 4  |-  A  e. 
_V
32alexeq 2897 . . 3  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
4 elsn 3655 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
54imbi1i 315 . . . 4  |-  ( ( x  e.  { A }  ->  ph )  <->  ( x  =  A  ->  ph )
)
65albii 1553 . . 3  |-  ( A. x ( x  e. 
{ A }  ->  ph )  <->  A. x ( x  =  A  ->  ph )
)
74anbi1i 676 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
87exbii 1569 . . 3  |-  ( E. x ( x  e. 
{ A }  /\  ph )  <->  E. x ( x  =  A  /\  ph ) )
93, 6, 83bitr4i 268 . 2  |-  ( A. x ( x  e. 
{ A }  ->  ph )  <->  E. x ( x  e.  { A }  /\  ph ) )
10 df-ral 2548 . 2  |-  ( A. x  e.  { A } ph  <->  A. x ( x  e.  { A }  ->  ph ) )
11 df-rex 2549 . 2  |-  ( E. x  e.  { A } ph  <->  E. x ( x  e.  { A }  /\  ph ) )
129, 10, 113bitr4i 268 1  |-  ( A. x  e.  { A } ph  <->  E. x  e.  { A } ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {csn 3640
This theorem is referenced by:  fatesg  24956
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rex 2549  df-v 2790  df-sn 3646
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