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Theorem dfiota2 4968
Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
dfiota2 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfiota2
StepHypRef Expression
1 df-iota 4967 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 df-sn 3447 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
32eqeq2i 2098 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
4 abbi 2201 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
53, 4bitr4i 185 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
65abbii 2203 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
76unieqi 3658 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
81, 7eqtri 2108 1 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287   = wceq 1289  {cab 2074  {csn 3441   cuni 3648  cio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-sn 3447  df-uni 3649  df-iota 4967
This theorem is referenced by:  nfiota1  4969  nfiotadxy  4970  cbviota  4972  sb8iota  4974  iotaval  4978  iotanul  4982  fv2  5284
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