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Theorem iotaexeu 39145
 Description: The iota class exists. This theorem does not require ax-nul 4923 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 6005 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
21eqcomd 2777 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
32eximi 1910 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4 df-eu 2622 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 isset 3359 . 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 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∃!weu 2618  Vcvv 3351  ℩cio 5992 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-sbc 3588  df-un 3728  df-sn 4317  df-pr 4319  df-uni 4575  df-iota 5994 This theorem is referenced by:  iotasbc  39146  pm14.18  39155  iotavalb  39157  sbiota1  39161
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