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Theorem sniota 5517
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 2275 . . 3  |-  F/ x E! x ph
2 iota1 5504 . . . . 5  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
3 eqcom 2463 . . . . 5  |-  ( ( iota x ph )  =  x  <->  x  =  ( iota x ph ) )
42, 3syl6bb 261 . . . 4  |-  ( E! x ph  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
5 abid 2441 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
6 vex 3081 . . . . 5  |-  x  e. 
_V
76elsnc 4010 . . . 4  |-  ( x  e.  { ( iota
x ph ) }  <->  x  =  ( iota x ph )
)
84, 5, 73bitr4g 288 . . 3  |-  ( E! x ph  ->  (
x  e.  { x  |  ph }  <->  x  e.  { ( iota x ph ) } ) )
91, 8alrimi 1816 . 2  |-  ( E! x ph  ->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
10 nfab1 2618 . . 3  |-  F/_ x { x  |  ph }
11 nfiota1 5492 . . . 4  |-  F/_ x
( iota x ph )
1211nfsn 4043 . . 3  |-  F/_ x { ( iota x ph ) }
1310, 12cleqf 2643 . 2  |-  ( { x  |  ph }  =  { ( iota x ph ) }  <->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
149, 13sylibr 212 1  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   E!weu 2262   {cab 2439   {csn 3986   iotacio 5488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-sbc 3295  df-un 3442  df-sn 3987  df-pr 3989  df-uni 4201  df-iota 5490
This theorem is referenced by:  snriota  6192
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