Theorem dfiota2 5090

Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
dfiota2	|- (iotaxph) = U.{y | A.x(ph <-> x = y)}

Distinct variable groups:   x,y   ph,y

Proof of Theorem dfiota2

Step	Hyp	Ref	Expression
1		df-iota 5089	. 2 |- (iotaxph) = U.{y | {x | ph} = {y}}
2		df-sn 3049	. . . . . 6 |- {y} = {x | x = y}
3	2	eqeq2i 1894	. . . . 5 |- ({x | ph} = {y} <-> {x | ph} = {x | x = y})
4		eq2ab 2004	. . . . 5 |- ({x | ph} = {x | x = y} <-> A.x(ph <-> x = y))
5	3, 4	bitri 190	. . . 4 |- ({x | ph} = {y} <-> A.x(ph <-> x = y))
6	5	abbii 2006	. . 3 |- {y | {x | ph} = {y}} = {y | A.x(ph <-> x = y)}
7	6	unieqi 3187	. 2 |- U.{y | {x | ph} = {y}} = U.{y | A.x(ph <-> x = y)}
8	1, 7	eqtri 1908	1 |- (iotaxph) = U.{y | A.x(ph <-> x = y)}

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298  {cab 1871  {csn 3044  U.cuni 3177  iotacio 5087

This theorem is referenced by:  iotaval 5096  iotanul 5098  fviota 16429  cbviotaf 16432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-sn 3049  df-uni 3178  df-iota 5089

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