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Theorem iotaexeu 38439
Description: The iota class exists. This theorem does not require ax-nul 4780 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaexeu (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Proof of Theorem iotaexeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iotaval 5850 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
21eqcomd 2626 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
32eximi 1760 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4 df-eu 2472 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 isset 3202 . 2 ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
63, 4, 53imtr4i 281 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479   = wceq 1481  wex 1702  wcel 1988  ∃!weu 2468  Vcvv 3195  cio 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-sbc 3430  df-un 3572  df-sn 4169  df-pr 4171  df-uni 4428  df-iota 5839
This theorem is referenced by:  iotasbc  38440  pm14.18  38449  iotavalb  38451  sbiota1  38455
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