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Theorem dfiota2 5220
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 5219 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 df-sn 3646 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
32eqeq2i 2293 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
x  |  x  =  y } )
4 abbi 2393 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
53, 4bitr4i 243 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
65abbii 2395 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { y  |  A. x ( ph  <->  x  =  y ) }
76unieqi 3837 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
81, 7eqtri 2303 1  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527    = wceq 1623   {cab 2269   {csn 3640   U.cuni 3827   iotacio 5217
This theorem is referenced by:  nfiota1  5221  nfiotad  5222  cbviota  5224  sb8iota  5226  iotaval  5230  iotanul  5234  fv2  5520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-sn 3646  df-uni 3828  df-iota 5219
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