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Theorem sniota 4994
Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
sniota  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )

Proof of Theorem sniota
StepHypRef Expression
1 nfeu1 1959 . . 3  |-  F/ x E! x ph
2 iota1 4981 . . . . 5  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
3 eqcom 2090 . . . . 5  |-  ( ( iota x ph )  =  x  <->  x  =  ( iota x ph ) )
42, 3syl6bb 194 . . . 4  |-  ( E! x ph  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
5 abid 2076 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
6 vex 2622 . . . . 5  |-  x  e. 
_V
76elsn 3457 . . . 4  |-  ( x  e.  { ( iota
x ph ) }  <->  x  =  ( iota x ph )
)
84, 5, 73bitr4g 221 . . 3  |-  ( E! x ph  ->  (
x  e.  { x  |  ph }  <->  x  e.  { ( iota x ph ) } ) )
91, 8alrimi 1460 . 2  |-  ( E! x ph  ->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
10 nfab1 2230 . . 3  |-  F/_ x { x  |  ph }
11 nfiota1 4969 . . . 4  |-  F/_ x
( iota x ph )
1211nfsn 3497 . . 3  |-  F/_ x { ( iota x ph ) }
1310, 12cleqf 2252 . 2  |-  ( { x  |  ph }  =  { ( iota x ph ) }  <->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
149, 13sylibr 132 1  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   E!weu 1948   {cab 2074   {csn 3441   iotacio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967
This theorem is referenced by:  snriota  5619
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