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Theorem sn-iotaex 40184
Description: iotaex 6408 without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
sn-iotaex (℩𝑥𝜑) ∈ V

Proof of Theorem sn-iotaex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iotavallem 40179 . . . 4 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2 vex 3435 . . . 4 𝑦 ∈ V
31, 2eqeltrdi 2847 . . 3 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
43exlimiv 1933 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5 sn-iotanul 40181 . . 3 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6 0ex 5231 . . 3 ∅ ∈ V
75, 6eqeltrdi 2847 . 2 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
84, 7pm2.61i 182 1 (℩𝑥𝜑) ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wex 1782  wcel 2106  {cab 2715  Vcvv 3431  c0 4258  {csn 4563  cio 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-sn 4564  df-pr 4566  df-uni 4842  df-iota 6386
This theorem is referenced by: (None)
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