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Theorem dfaiota2 45794
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 45793 . 2 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 absn 4647 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
32abbii 2803 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
43inteqi 4955 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
51, 4eqtri 2761 1 (℩'𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540   = wceq 1542  {cab 2710  {csn 4629   cint 4951  ℩'caiota 45791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-ral 3063  df-rex 3072  df-sn 4630  df-int 4952  df-aiota 45793
This theorem is referenced by: (None)
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