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Theorem iotaexOLD 6520
Description: Obsolete version of iotaex 6513 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
iotaexOLD (℩𝑥𝜑) ∈ V

Proof of Theorem iotaexOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 iotaval 6511 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
21eqcomd 2739 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
32eximi 1838 . . 3 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4 eu6 2569 . . 3 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
5 isset 3488 . . 3 ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
63, 4, 53imtr4i 292 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7 iotanul 6518 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8 0ex 5306 . . 3 ∅ ∈ V
97, 8eqeltrdi 2842 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
106, 9pm2.61i 182 1 (℩𝑥𝜑) ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2563  Vcvv 3475  c0 4321  cio 6490
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6492
This theorem is referenced by: (None)
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