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Theorem dfiota2 4795
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 4794 . 2 (℩xφ) = {y ∣ {xφ} = {y}}
2 df-sn 3356 . . . . . 6 {y} = {xx = y}
32eqeq2i 2032 . . . . 5 ({xφ} = {y} ↔ {xφ} = {xx = y})
4 abbi 2133 . . . . 5 (x(φx = y) ↔ {xφ} = {xx = y})
53, 4bitr4i 176 . . . 4 ({xφ} = {y} ↔ x(φx = y))
65abbii 2135 . . 3 {y ∣ {xφ} = {y}} = {yx(φx = y)}
76unieqi 3564 . 2 {y ∣ {xφ} = {y}} = {yx(φx = y)}
81, 7eqtri 2042 1 (℩xφ) = {yx(φx = y)}
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1226   = wceq 1228  {cab 2008  {csn 3350   cuni 3554  cio 4792
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-sn 3356  df-uni 3555  df-iota 4794
This theorem is referenced by:  nfiota1  4796  nfiotadxy  4797  cbviota  4799  sb8iota  4801  iotaval  4805  iotanul  4809  fv2  5098
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