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Theorem iotauni 5370
Description: Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iotauni  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )

Proof of Theorem iotauni
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2242 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 iotaval 5369 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
3 uniabio 5368 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  U. {
x  |  ph }  =  z )
42, 3eqtr4d 2422 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  = 
U. { x  | 
ph } )
54exlimiv 1641 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( iota x ph )  =  U. { x  |  ph }
)
61, 5sylbi 188 1  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649   E!weu 2238   {cab 2373   U.cuni 3957   iotacio 5356
This theorem is referenced by:  iotaint  5371  iotassuni  5374  dfiota4  5386  fveu  5660  riotauni  6492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-v 2901  df-sbc 3105  df-un 3268  df-sn 3763  df-pr 3764  df-uni 3958  df-iota 5358
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