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Theorem iotaexeu 40612
Description: The iota class exists. This theorem does not require ax-nul 5206 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 6326 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
21eqcomd 2830 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
32eximi 1828 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4 eu6 2654 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 isset 3511 . 2 ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
63, 4, 53imtr4i 293 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528   = wceq 1530  wex 1773  wcel 2106  ∃!weu 2648  Vcvv 3499  cio 6309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rex 3148  df-v 3501  df-sbc 3776  df-un 3944  df-sn 4564  df-pr 4566  df-uni 4837  df-iota 6311
This theorem is referenced by:  iotasbc  40613  pm14.18  40622  iotavalb  40624  sbiota1  40628
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