Theorem dfiota2 5197

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

Step	Hyp	Ref	Expression
1		df-iota 5196	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		df-sn 3613	. . . . . 6 |- { y } = { x | x = y }
3	2	eqeq2i 2200	. . . . 5 |- ( { x | ph } = { y } <-> { x | ph } = { x | x = y } )
4		abbi 2303	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
5	3, 4	bitr4i 187	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
6	5	abbii 2305	. . 3 |- { y | { x | ph } = { y } } = { y | A. x ( ph <-> x = y ) }
7	6	unieqi 3834	. 2 |- U. { y | { x | ph } = { y } } = U. { y | A. x ( ph <-> x = y ) }
8	1, 7	eqtri 2210	1 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   {cab 2175   {csn 3607   U.cuni 3824   iotacio 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-sn 3613  df-uni 3825  df-iota 5196

This theorem is referenced by:  nfiota1  5198  nfiotadw  5199  cbviota  5201  sb8iota  5203  iotaval  5207  iotanul  5211  eliota  5223  fv2  5529