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Theorem reuabaiotaiota 47036
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 4990 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
2 df-iota 6515 . . 3 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
3 df-aiota 47034 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
42, 3eqeq12i 2752 . 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 1536  ∃!weu 2565  {cab 2711  {csn 4630   cuni 4911   cint 4950  cio 6513  ℩'caiota 47032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-uni 4912  df-int 4951  df-iota 6515  df-aiota 47034
This theorem is referenced by:  reuaiotaiota  47037  aiotaval  47044
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