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Theorem reuabaiotaiota 43294
 Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥 ∣ 𝜑} = {𝑦}. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
reuabaiotaiota (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuabaiotaiota
StepHypRef Expression
1 uniintab 4917 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
2 df-iota 6317 . . 3 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
3 df-aiota 43292 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
42, 3eqeq12i 2839 . 2 ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
51, 4bitr4i 280 1 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1536  ∃!weu 2652  {cab 2802  {csn 4570  ∪ cuni 4841  ∩ cint 4879  ℩cio 6315  ℩'caiota 43290 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-sn 4571  df-pr 4573  df-uni 4842  df-int 4880  df-iota 6317  df-aiota 43292 This theorem is referenced by:  reuaiotaiota  43295  aiotaval  43300
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