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Theorem reuabaiotaiota 45785
Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥𝜑} = {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 4992 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
2 df-iota 6495 . . 3 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
3 df-aiota 45783 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
42, 3eqeq12i 2750 . 2 ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
51, 4bitr4i 277 1 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  ∃!weu 2562  {cab 2709  {csn 4628   cuni 4908   cint 4950  cio 6493  ℩'caiota 45781
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-int 4951  df-iota 6495  df-aiota 45783
This theorem is referenced by:  reuaiotaiota  45786  aiotaval  45793
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