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Theorem reuabaiotaiota 47099
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 4986 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
2 df-iota 6514 . . 3 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
3 df-aiota 47097 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
42, 3eqeq12i 2755 . 2 ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
51, 4bitr4i 278 1 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  ∃!weu 2568  {cab 2714  {csn 4626   cuni 4907   cint 4946  cio 6512  ℩'caiota 47095
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-int 4947  df-iota 6514  df-aiota 47097
This theorem is referenced by:  reuaiotaiota  47100  aiotaval  47107
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