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Theorem iotaex 5832
Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaex (℩𝑥𝜑) ∈ V

Proof of Theorem iotaex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 iotaval 5826 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
21eqcomd 2627 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
32eximi 1759 . . 3 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4 df-eu 2473 . . 3 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
5 isset 3196 . . 3 ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
63, 4, 53imtr4i 281 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7 iotanul 5830 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8 0ex 4755 . . 3 ∅ ∈ V
97, 8syl6eqel 2706 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
106, 9pm2.61i 176 1 (℩𝑥𝜑) ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1478   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  Vcvv 3189  c0 3896  cio 5813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5815
This theorem is referenced by:  iota4an  5834  fvex  6163  riotaex  6575  erov  7796  iunfictbso  8889  isf32lem9  9135  sumex  14360  prodex  14573  pcval  15484  grpidval  17192  fn0g  17194  gsumvalx  17202  psgnfn  17853  psgnval  17859  dchrptlem1  24906  lgsdchrval  24996  lgsdchr  24997  bnj1366  30643  nosino  31610  nosifv  31611  nosires  31612  bj-finsumval0  32815  ellimciota  39278  fourierdlem36  39693
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