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Theorem dfiota2 5045
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 5044 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 df-sn 3497 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
32eqeq2i 2123 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
4 abbi 2226 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
53, 4bitr4i 186 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
65abbii 2228 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
76unieqi 3710 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
81, 7eqtri 2133 1 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1310   = wceq 1312  {cab 2099  {csn 3491   cuni 3700  cio 5042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-sn 3497  df-uni 3701  df-iota 5044
This theorem is referenced by:  nfiota1  5046  nfiotadxy  5047  cbviota  5049  sb8iota  5051  iotaval  5055  iotanul  5059  fv2  5368
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