Theorem dfiota2 4341

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

Allowed substitution hint:   ph(x)

Proof of Theorem dfiota2

Step	Hyp	Ref	Expression
1		df-iota 4340	. 2 |- (iotaxph) = U.{y | {x | ph} = {y}}
2		df-sn 3742	. . . . . 6 |- {y} = {x | x = y}
3	2	eqeq2i 2363	. . . . 5 |- ({x | ph} = {y} <-> {x | ph} = {x | x = y})
4		abbi 2464	. . . . 5 |- (A.x(ph <-> x = y) <-> {x | ph} = {x | x = y})
5	3, 4	bitr4i 243	. . . 4 |- ({x | ph} = {y} <-> A.x(ph <-> x = y))
6	5	abbii 2466	. . 3 |- {y | {x | ph} = {y}} = {y | A.x(ph <-> x = y)}
7	6	unieqi 3902	. 2 |- U.{y | {x | ph} = {y}} = U.{y | A.x(ph <-> x = y)}
8	1, 7	eqtri 2373	1 |- (iotaxph) = U.{y | A.x(ph <-> x = y)}

Syntax hints:   <-> wb 176  A.wal 1540   = wceq 1642  {cab 2339  {csn 3738  U.cuni 3892  iotacio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  nfiota1  4342  nfiotad  4343  cbviota  4345  sb8iota  4347  iotaval  4351  iotanul  4355  eqtfinrelk  4487  fv2  5325  tcfnex  6245