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Theorem sniota 5246
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 2153 . . 3  |-  F/ x E! x ph
2 iota1 5233 . . . . 5  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
3 eqcom 2285 . . . . 5  |-  ( ( iota x ph )  =  x  <->  x  =  ( iota x ph ) )
42, 3syl6bb 252 . . . 4  |-  ( E! x ph  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
5 abid 2271 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
6 vex 2791 . . . . 5  |-  x  e. 
_V
76elsnc 3663 . . . 4  |-  ( x  e.  { ( iota
x ph ) }  <->  x  =  ( iota x ph )
)
84, 5, 73bitr4g 279 . . 3  |-  ( E! x ph  ->  (
x  e.  { x  |  ph }  <->  x  e.  { ( iota x ph ) } ) )
91, 8alrimi 1745 . 2  |-  ( E! x ph  ->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
10 nfab1 2421 . . 3  |-  F/_ x { x  |  ph }
11 nfiota1 5221 . . . 4  |-  F/_ x
( iota x ph )
1211nfsn 3691 . . 3  |-  F/_ x { ( iota x ph ) }
1310, 12cleqf 2443 . 2  |-  ( { x  |  ph }  =  { ( iota x ph ) }  <->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
149, 13sylibr 203 1  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   {csn 3640   iotacio 5217
This theorem is referenced by:  snriota  6335
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-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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