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Theorem dfiota2 4994
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 4993 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 df-sn 3456 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
32eqeq2i 2099 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
x  |  x  =  y } )
4 abbi 2202 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
53, 4bitr4i 186 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
65abbii 2204 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { y  |  A. x ( ph  <->  x  =  y ) }
76unieqi 3669 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
81, 7eqtri 2109 1  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1288    = wceq 1290   {cab 2075   {csn 3450   U.cuni 3659   iotacio 4991
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-sn 3456  df-uni 3660  df-iota 4993
This theorem is referenced by:  nfiota1  4995  nfiotadxy  4996  cbviota  4998  sb8iota  5000  iotaval  5004  iotanul  5008  fv2  5313
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