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Theorem iotaexeu 41495
 Description: The iota class exists. This theorem does not require ax-nul 5176 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 6309 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
21eqcomd 2764 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
32eximi 1836 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4 eu6 2593 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 isset 3422 . 2 ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
63, 4, 53imtr4i 295 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2587  Vcvv 3409  ℩cio 6292 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-sbc 3697  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-uni 4799  df-iota 6294 This theorem is referenced by:  iotasbc  41496  pm14.18  41505  iotavalb  41507  sbiota1  41511
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