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Theorem aiotaint 47103
Description: This is to df-aiota 47097 what iotauni 6536 is to df-iota 6514 (it uses intersection like df-aiota 47097, similar to iotauni 6536 using union like df-iota 6514; we could also prove an analogous result using union here too, in the same way that we have iotaint 6537). (Contributed by BJ, 31-Aug-2024.)
Assertion
Ref Expression
aiotaint (∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑})

Proof of Theorem aiotaint
StepHypRef Expression
1 reuaiotaiota 47100 . . 3 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
21biimpi 216 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = (℩'𝑥𝜑))
3 iotaint 6537 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
42, 3eqtr3d 2779 1 (∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ∃!weu 2568  {cab 2714   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:  dfaiota3  47104
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