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Theorem sniota 5584
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 2295 . 2  |-  F/ x E! x ph
2 nfab1 2621 . 2  |-  F/_ x { x  |  ph }
3 nfiota1 5559 . . 3  |-  F/_ x
( iota x ph )
43nfsn 4089 . 2  |-  F/_ x { ( iota x ph ) }
5 iota1 5571 . . . 4  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
6 eqcom 2466 . . . 4  |-  ( ( iota x ph )  =  x  <->  x  =  ( iota x ph ) )
75, 6syl6bb 261 . . 3  |-  ( E! x ph  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
8 abid 2444 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
9 vex 3112 . . . 4  |-  x  e. 
_V
109elsnc 4056 . . 3  |-  ( x  e.  { ( iota
x ph ) }  <->  x  =  ( iota x ph )
)
117, 8, 103bitr4g 288 . 2  |-  ( E! x ph  ->  (
x  e.  { x  |  ph }  <->  x  e.  { ( iota x ph ) } ) )
121, 2, 4, 11eqrd 3517 1  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   E!weu 2283   {cab 2442   {csn 4032   iotacio 5555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-un 3476  df-in 3478  df-ss 3485  df-sn 4033  df-pr 4035  df-uni 4252  df-iota 5557
This theorem is referenced by:  snriota  6287
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