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Theorem dfiota2 5509
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 5508 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 df-sn 3942 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
32eqeq2i 2440 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
x  |  x  =  y } )
4 abbi 2542 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
53, 4bitr4i 255 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
65abbii 2544 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { y  |  A. x ( ph  <->  x  =  y ) }
76unieqi 4171 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
81, 7eqtri 2450 1  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435    = wceq 1437   {cab 2414   {csn 3941   U.cuni 4162   iotacio 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rex 2720  df-sn 3942  df-uni 4163  df-iota 5508
This theorem is referenced by:  nfiota1  5510  nfiotad  5511  cbviota  5513  sb8iota  5515  iotaval  5519  iotanul  5523  fv2  5820
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