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Theorem fates 25058
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 2810 . . . 4  |-  A  e. 
_V
32alexeq 2910 . . 3  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
4 elsn 3668 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
54imbi1i 315 . . . 4  |-  ( ( x  e.  { A }  ->  ph )  <->  ( x  =  A  ->  ph )
)
65albii 1556 . . 3  |-  ( A. x ( x  e. 
{ A }  ->  ph )  <->  A. x ( x  =  A  ->  ph )
)
74anbi1i 676 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( x  =  A  /\  ph )
)
87exbii 1572 . . 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 2561 . 2  |-  ( A. x  e.  { A } ph  <->  A. x ( x  e.  { A }  ->  ph ) )
11 df-rex 2562 . 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 1530   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {csn 3653
This theorem is referenced by:  fatesg  25059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-rex 2562  df-v 2803  df-sn 3659
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