Theorem aiotaexb 47552

Description: The alternate iota over a wff 𝜑 is a set iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
aiotaexb	⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Proof of Theorem aiotaexb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intexab 5274	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
2		euabsn2 4657	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3		df-aiota 47548	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	3	eleq1i 2830	. 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	1, 2, 4	3bitr4i 304	1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Syntax hints:   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572  {cab 2717  Vcvv 3431  {csn 4555  ∩ cint 4877  ℩'caiota 47546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4262  df-sn 4556  df-int 4878  df-aiota 47548

This theorem is referenced by:  aiotavb  47553  iotan0aiotaex  47556  aiotaexaiotaiota  47557  aiota0ndef  47560