Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iotaexeu Structured version   Unicode version

Theorem iotaexeu 36739
Description: The iota class exists. This theorem does not require ax-nul 4555 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 5576 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
21eqcomd 2430 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
32eximi 1701 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y  y  =  ( iota x ph ) )
4 df-eu 2273 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
5 isset 3084 . 2  |-  ( ( iota x ph )  e.  _V  <->  E. y  y  =  ( iota x ph ) )
63, 4, 53imtr4i 269 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   E!weu 2269   _Vcvv 3080   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:  iotasbc  36740  pm14.18  36749  iotavalb  36751  sbiota1  36755
  Copyright terms: Public domain W3C validator