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Theorem aiotaexaiotaiota 45802
Description: The alternate iota over a wff 𝜑 is a set iff the iota and the alternate iota over 𝜑 are equal. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiotaexaiotaiota ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem aiotaexaiotaiota
StepHypRef Expression
1 aiotaexb 45797 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
2 reuaiotaiota 45796 . 2 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitr3i 277 1 ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  ∃!weu 2563  Vcvv 3475  cio 6494  ℩'caiota 45791
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-sep 5300
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-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4910  df-int 4952  df-iota 6496  df-aiota 45793
This theorem is referenced by: (None)
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