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Theorem dfaiota2 44037
 Description: Alternate definition of the alternate version of Russell's definition of a description binder. Definition 8.18 in [Quine] p. 56. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
dfaiota2 (℩'𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfaiota2
StepHypRef Expression
1 df-aiota 44036 . 2 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 absn 4543 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
32abbii 2823 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
43inteqi 4845 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
51, 4eqtri 2781 1 (℩'𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538  {cab 2735  {csn 4525  ∩ cint 4841  ℩'caiota 44034 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-sn 4526  df-int 4842  df-aiota 44036 This theorem is referenced by: (None)
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