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Theorem dfiota2 4868
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 4867 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 df-sn 3381 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
32eqeq2i 2050 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
4 abbi 2151 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
53, 4bitr4i 176 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
65abbii 2153 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
76unieqi 3590 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
81, 7eqtri 2060 1 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1241   = wceq 1243  {cab 2026  {csn 3375   cuni 3580  cio 4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-sn 3381  df-uni 3581  df-iota 4867
This theorem is referenced by:  nfiota1  4869  nfiotadxy  4870  cbviota  4872  sb8iota  4874  iotaval  4878  iotanul  4882  fv2  5173
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