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Theorem dfiota2 5410
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 5409 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 df-sn 3812 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
32eqeq2i 2445 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  {
x  |  x  =  y } )
4 abbi 2545 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  <->  { x  |  ph }  =  {
x  |  x  =  y } )
53, 4bitr4i 244 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
65abbii 2547 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  { y  |  A. x ( ph  <->  x  =  y ) }
76unieqi 4017 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
81, 7eqtri 2455 1  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549    = wceq 1652   {cab 2421   {csn 3806   U.cuni 4007   iotacio 5407
This theorem is referenced by:  nfiota1  5411  nfiotad  5412  cbviota  5414  sb8iota  5416  iotaval  5420  iotanul  5424  fv2  5714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-sn 3812  df-uni 4008  df-iota 5409
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