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Theorem dfiota2 4981
Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
dfiota2  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem dfiota2
StepHypRef Expression
1 df-iota 4980 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 df-sn 3452 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
32eqeq2i 2098 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
x  |  x  =  y } )
4 abbi 2201 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
53, 4bitr4i 185 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
65abbii 2203 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { y  |  A. x ( ph  <->  x  =  y ) }
76unieqi 3663 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
81, 7eqtri 2108 1  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287    = wceq 1289   {cab 2074   {csn 3446   U.cuni 3653   iotacio 4978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-sn 3452  df-uni 3654  df-iota 4980
This theorem is referenced by:  nfiota1  4982  nfiotadxy  4983  cbviota  4985  sb8iota  4987  iotaval  4991  iotanul  4995  fv2  5300
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