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Theorem iotaexeu 27280
Description: The iota class exists. This theorem does not require ax-nul 4272 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaexeu  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )

Proof of Theorem iotaexeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iotaval 5362 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
21eqcomd 2385 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
32eximi 1582 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y  y  =  ( iota x ph ) )
4 df-eu 2235 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
5 isset 2896 . 2  |-  ( ( iota x ph )  e.  _V  <->  E. y  y  =  ( iota x ph ) )
63, 4, 53imtr4i 258 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2231   _Vcvv 2892   iotacio 5349
This theorem is referenced by:  iotasbc  27281  pm14.18  27290  iotavalb  27292  sbiota1  27296
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 2361
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 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-v 2894  df-sbc 3098  df-un 3261  df-sn 3756  df-pr 3757  df-uni 3951  df-iota 5351
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