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Theorem iotauni 5577
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 2273 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 iotaval 5576 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
3 uniabio 5575 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  U. {
x  |  ph }  =  z )
42, 3eqtr4d 2466 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  = 
U. { x  | 
ph } )
54exlimiv 1770 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( iota x ph )  =  U. { x  |  ph }
)
61, 5sylbi 198 1  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1657   E!weu 2269   {cab 2407   U.cuni 4219   iotacio 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565
This theorem is referenced by:  iotaint  5578  iotassuni  5581  dfiota4  5593  fveu  5873  riotauni  6273
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