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Theorem dfiota2 4811
Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
dfiota2 (℩xφ) = {yx(φx = y)}
Distinct variable groups:   x,y   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem dfiota2
StepHypRef Expression
1 df-iota 4810 . 2 (℩xφ) = {y ∣ {xφ} = {y}}
2 df-sn 3373 . . . . . 6 {y} = {xx = y}
32eqeq2i 2047 . . . . 5 ({xφ} = {y} ↔ {xφ} = {xx = y})
4 abbi 2148 . . . . 5 (x(φx = y) ↔ {xφ} = {xx = y})
53, 4bitr4i 176 . . . 4 ({xφ} = {y} ↔ x(φx = y))
65abbii 2150 . . 3 {y ∣ {xφ} = {y}} = {yx(φx = y)}
76unieqi 3581 . 2 {y ∣ {xφ} = {y}} = {yx(φx = y)}
81, 7eqtri 2057 1 (℩xφ) = {yx(φx = y)}
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242  {cab 2023  {csn 3367   cuni 3571  cio 4808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sn 3373  df-uni 3572  df-iota 4810
This theorem is referenced by:  nfiota1  4812  nfiotadxy  4813  cbviota  4815  sb8iota  4817  iotaval  4821  iotanul  4825  fv2  5116
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