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Theorem dfiota2 5750
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 5749 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 df-sn 4120 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
32eqeq2i 2616 . . . . 5 ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
4 abbi 2718 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑦})
53, 4bitr4i 265 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
65abbii 2720 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
76unieqi 4370 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
81, 7eqtri 2626 1 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 194  wal 1472   = wceq 1474  {cab 2590  {csn 4119   cuni 4361  cio 5747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-rex 2896  df-sn 4120  df-uni 4362  df-iota 5749
This theorem is referenced by:  nfiota1  5751  nfiotad  5752  cbviota  5754  sb8iota  5756  iotaval  5760  iotanul  5764  fv2  6078
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