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Theorem iotaex 5368
Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the  iota class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaex  |-  ( iota
x ph )  e.  _V

Proof of Theorem iotaex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 iotaval 5362 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2385 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1582 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2235 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 2896 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 258 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 5366 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4273 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2468 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  e.  _V )
106, 9pm2.61i 158 1  |-  ( iota
x ph )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2231   _Vcvv 2892   (/)c0 3564   iotacio 5349
This theorem is referenced by:  iota4an  5370  fvex  5675  riotaex  6482  erov  6930  iunfictbso  7921  isf32lem9  8167  sumex  12401  pcval  13138  grpidval  14627  fn0g  14628  gsumvalx  14694  dchrptlem1  20908  lgsdchrval  20991  lgsdchr  20992  prodex  25005  psgnfn  27086  psgnval  27092  bnj1366  28532
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  ax-nul 4272
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-ne 2545  df-ral 2647  df-rex 2648  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-sn 3756  df-pr 3757  df-uni 3951  df-iota 5351
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